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Let’s determine an expression that will allow us to calculate the rate of this energy transfer. First, consider the simple circuit in Figure (--), where energy is delivered to a resistor. Because the connecting wires also have resistance, some energy is delivered to the wires and some to the resistor. Unless noted otherwise, we shall assume the resistance of the wires is small compared with the resistance of the circuit element so that the energy delivered to the wires is negligible. Imagine following a positive quantity of charge Q moving clockwise around the circuit from point a through the battery and resistor back to point a. We identify the entire circuit as our system. As the charge moves from a to b through the battery, the electric potential energy of the system increases by an amount while the chemical potential energy in the battery decreases by the same amount. As the charge moves from c to d through the resistor, however, the electric potential energy of the system decreases due to collisions of electrons with atoms in the resistor. In this process, the electric potential energy is transformed to internal energy corresponding to increased vibrational motion of the atoms in the resistor. Because the resistance of the interconnecting wires is neglected, no energy transformation occurs for paths bc and da. When the charge returns to point a, the net result is that some of the chemical energy in the battery has been delivered to the resistor and resides in the resistor as internal energy associated with molecular vibration. The resistor is normally in contact with air, so its increased temperature results in a transfer of energy by heat into the air. In addition, the resistor emits thermal radiation, representing another means of escape for the energy. After some time interval has passed, the resistor reaches a constant temperature. At this time, the input of energy from the battery is balanced by the output of energy from the resistor by heat and radiation. Some electrical devices include heat sinks3 connected to parts of the circuit to prevent these parts from reaching dangerously high temperatures. Heat sinks are pieces of metal with many fins. Because the metal’s high thermal conductivity provides a rapid transfer of energy by heat away from the hot component and the large number of fins provides a large surface area in contact with the air, energy can transfer by radiation and into the air by heat at a high rate. I

Let’s now investigate the rate at which the electric potential energy of the system decreases as the charge Q passes through the resistor: () Where I is the current in the circuit. The system regains this potential energy when the charge passes through the battery, at the expense of chemical energy in the battery. The rate at which the potential energy of the system decreases as the charge passes through the resistor is equal to the rate at which the system gains internal energy in the resistor. Therefore, the power P, representing the rate at which energy is delivered to the resistor, is We derived this result by considering a battery delivering energy to a resistor. Equation (--), however, can be used to calculate the power delivered by a voltage source to any device carrying a current I and having a potential difference V between its terminals. To calculate the electric power density (power per unit volume), suppose a closed surface in cylinder form of section area S and length L, the area of closed surface is A encloses a volume V=SL, Then

() () Power density is scalar quantity, therefore the product between current density vector J and electric field vector E is scalar product. The power density is proportion to the square of electric field.

Magnetostatic field: Gauss’s law of magnetostatics Several experiments showed that every natural permanent magnet, regardless of its shape, has two poles, called north (N) and south (S) poles, that exert forces on other magnetic poles similar to the way electric charges exert forces on one another. That is, like poles (N–N or S–S) repel each other, and opposite poles (N– S) attract each other. The poles received their names because of the way a magnet, such as that in a compass, behaves in the presence of the Earth’s magnetic field. If a bar magnet is suspended from its midpoint and can swing freely in a horizontal plane, it will rotate until its north pole points to the Earth’s geographic North Pole and its south pole points to the Earth’s geographic South Pole. The Earth itself is a large, permanent magnet. In 1750, experimenters used a torsion balance to show that magnetic poles exert attractive or repulsive forces on each other and that these forces vary as the inverse square of the distance between interacting poles. Although the force between two magnetic poles is otherwise similar to the force between two electric charges, electric charges can be isolated (witness the electron and proton), whereas a single magnetic pole has never been isolated. That is, magnetic poles are always found in pairs. All attempts thus far to detect an isolated magnetic pole have been unsuccessful. No matter how many times a permanent magnet is cut in two, each piece always has a north and a south pole The relationship between magnetism and electricity was discovered in 1819 when, during a lecture demonstration, Hans Christian Oersted found that an electric current in a wire deflected a nearby compass needle. In the 1820s, further connections between electricity and magnetism were demonstrated independently by Faraday and Joseph Henry (1797–1878). They showed that an electric current can be produced in a circuit either by moving a magnet near the circuit or by changing the current in a nearby circuit. These observations demonstrate that a changing magnetic field creates an electric field. Years later, theoretical work by Maxwell showed that the reverse is also true: a changing electric field creates a magnetic field.

Magnetic force As discussed early, the electric force is appear between two point charges as represented by Coulomb’s law From this equation it is found that the electric force is dependence on the electric charges and square distance between them. It is obey the inverse square law, with appropriate proportional constant ke. The question arise here is can it expressed the magnetic force with the same manner of electric force?, i.e. with regarding the magnetic force is exist between the two poles P1 (North pole) and P2 (South pole) and obey the inverse square law. If it is true, the magnetic force is given by This equation is incorrect because the separation distance r cannot reach infinity, while in electric force the distant may be reach infinity. When the distance r reach to infinity, the two charges could be separated to obtain a single electric charge, while naturally a single pole of permanent magnet impossible to obtained, why? When a permanent magnet is broken into two pieces to try separation of the two poles, two permanent magnets were obtained.

Therefore the magnetic force was expressed in different manner as follows. Experimentally it is found that the magnetic force arise only in the case when test point electric charge moves inside the magnetic field. The magnetic field B can defined at some point in space in terms of the magnetic force FB the field exerts on a charged particle moving with a velocity v , which we call the test charge. For the time being, let’s assume no electric or gravitational fields are present at the location of the test charge. Experiments on various charged particles moving in a magnetic field give the following results: 1- The magnitude FB of the magnetic force exerted on the particle is proportional to the charge q and to the speed v of the particle. 2- When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero. 3- When the particle’s velocity vector makes any angle  0 with the magnetic field, the magnetic force acts in a direction perpendicular to both v and B, that is, FB is perpendicular to the plane formed by v and B as in fig (a). 4- The magnetic force exerted on a positive charge is in the direction opposite the direction of the magnetic force exerted on a negative charge moving in the same direction as in fig (b). 5- The magnitude of the magnetic force exerted on the moving particle is proportional to sin, where  is the angle the particle’s velocity vector v makes with the direction of B

We can summarize these observations by writing the magnetic force in the form ⃗⃗⃗⃗ ⃗ ⃗ which by definition of the cross product is perpendicular to both v and B. We can regard this equation as an operational definition of the magnetic field at some point in space. That is, the magnetic field is defined in terms of the force acting on a moving charged particle. Direction of magnetic field Figure 29.5 reviews two right-hand rules for determining the direction of the cross product vxB and determining the direction of FB. The rule in Figure 29.5a depends on our right-hand rule for the cross product in Figure 11.2. Point the four fingers of your right hand along the direction of v with the palm facing B and curl them toward B S . Your extended thumb, which is at a right angle to your fingers, points in the direction of vxB. Because F=qvxB, FB is in the direction of your thumb if q is positive and is opposite the direction of your thumb if q is negative. An alternative rule is shown in Figure 29.5b. Here the thumb points in the direction of v and the extended fingers in the direction of B . Now, the force FB on a positive charge extends outward from the palm. The advantage of this rule is that the force on the charge is in the direction you would push on something with your

hand: outward from your palm. The force on a negative charge is in the opposite direction. You can use either of these two right-hand rules. The magnitude of the magnetic force on a charged particle is The angle  is the smaller angle between v and B. From this expression, we see that FB is zero when v is parallel or antiparallel to B (=0 0r 180) and maximum when v is perpendicular to B (=90). Magnetic field of natural permanent magnet In the previous section, the magnetic force represented in different form electric force, thus the magnetic field should be expressed in different form of electric field. Can the magnetic field B expressed in the same manner of electric field? The electric field or in general, the field is the force zone or influence region in which the force was affect on any test particle inside this region. The electric field is defined as the region around the charge in which the electric force is effect, force per unit charge; In the same manner, the magnetic field can be express as a magnetic force per unit pole; That is incorrect because there is no unit pole. Or can the magnetic field expressed by That is also incorrect because in some moment the magnetic field equals zero, but the magnetic field is exist always when also the moving charge does not exist. Mathematically the magnetic field can expressed by studying the Biot-Savart law in the next section.

In our study of electricity, we described the interactions between charged objects in terms of electric fields. Recall that an electric field surrounds any electric charge. In addition to containing an electric field, the region of space surrounding any moving electric charge also contains a magnetic field. A magnetic field also surrounds a magnetic substance making up a permanent magnet. Historically, the symbol B has been used to represent a magnetic field, and we use this notation in this book. The direction of the magnetic field B at any location is the direction in which a compass needle points at that location. As with the electric field, we can represent the magnetic field by means of drawings with magnetic field lines. Figure (--) shows how the magnetic field lines of a bar magnet can be traced with the aid of a compass. Notice that the magnetic field lines outside the magnet point away from the north pole and toward the south pole. One can display magnetic field patterns of a bar magnet using small iron filings.

From this section it is noted that the known source of magnetic field is natural permanent magnet, there are other sources discussed later. Magnetic flux and Gauss’s law in magnetostatics The flux associated with a magnetic field is defined in a manner similar to that used to define electric flux. Consider a closed surface A encloses a volume V. Consider an element of area (area vector) dA on an arbitrarily shaped closed surface as shown in Fig (--). If the magnetic field at this element is B , the magnetic flux through the element is ∫ , where dA is a vector that is perpendicular to the surface (area vector) and has a magnitude equal to the area dA. Therefore, the total magnetic flux through the surface is The angle  is the angle between the area vector dA and magnetic field vector B. If the magnetic field is parallel to the plane as in Figure (--), then =90 and the flux through the plane is zero. If the field is perpendicular to the plane as in Figure (--), then =0 and the flux through the plane is BA (the maximum value).

Closed surface Previously, we found that the electric flux through a closed surface surrounding a net charge is proportional to that charge (Gauss’s law). In other words, the number of electric field lines leaving the surface depends only on the net charge within it. This behavior exists because electric field lines originate and terminate on electric charges. The situation is quite different for magnetic fields, which are continuous and form closed loops. In other words, as illustrated by the magnetic field lines of a bar magnet, magnetic field lines do not begin or end at any point. For any closed surface such as the one outlined by the dashed line in Figure (--),

The number of lines entering the surface equals the number leaving the surface; therefore, the net magnetic flux is zero. In contrast, for a closed surface surrounding one charge of an electric dipole (Fig. (--)), the net electric flux is not zero. Gauss’s law in magnetism states that the net magnetic flux through any closed surface is always zero, mathematically ∮ This statement represents that isolated magnetic poles (monopoles) have never been detected and perhaps do not exist. Nonetheless, scientists continue the search because certain theories that are otherwise successful in explaining fundamental physical behavior suggest the possible existence of magnetic monopoles. Differences between electric field and magnetic field Electric and magnetic forces have several important differences: 1- The electric force vector is along the direction of the electric field, whereas the magnetic force vector is perpendicular to the magnetic field. 2- The electric force acts on a charged particle regardless of whether the particle is moving, whereas the magnetic force acts on a charged particle only when the particle is in motion. 3- The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a particle is displaced because the force is perpendicular to the displacement of its point of application.

Electrodynamics Ampere-Maxwell law The known natural source of magnetic field is the permanent magnet, but there are other two sources of the magnetic field which are passing the current through a wire (Biot-Savart law) and changing the electric field with time through open surface (Ampere-Maxwell law). In this chapter the Biot-Savart law and Ampere- Maxwell law would be discussed to derive the fourth equation of Maxwell. This chapter explores the origin of the magnetic field, moving charges. We begin by showing how to use the law of Biot and Savart to calculate the magnetic field produced at some point in space by a small current element. This formalism is then used to calculate the total magnetic field due to various current distributions. We also introduce Ampère’s law, which is useful in calculating the magnetic field of a highly symmetric configuration carrying a steady current. Magnetic field due to current carrying conductor wire: The Boit-Savart law Shortly after Oersted’s discovery in 1819 that a compass needle is deflected by a current-carrying conductor, Jean-Baptiste Biot (1774–1862) and Félix Savart (1791–1841) performed quantitative experiments on the force exerted by an electric current on a nearby magnet. From their experimental results, Biot and Savart arrived at a mathematical expression that gives the magnetic field at some point in space in terms of the current that produces the field. That expression is based on the following experimental observations for the magnetic field dB at a point P associated with a length element (displacement vector) dl of a wire carrying a steady current I as in Fig. (--)

1- The magnetic field vector dB is perpendicular both to dl (which points in the direction of the current) and to the unit vector r directed from dl toward P. 2- The magnitude of dB is inversely proportional to r2, where r is the distance from dl to P. 3- The magnitude of dB is proportional to the current and to the magnitude dl of the length element. 4- The magnitude of dB is proportional to sin , where  is the angle between the vectors dl and r . These observations are summarized in the mathematical expression known today as the Biot–Savart law: ⃗⃗⃗⃗⃗ ⃗⃗⃗ ̂ where o is a constant called the permeability of free space:

⁄ Notice that the field dB in Equation (--) is the field created at a point by the current in only a small length element dl of the conductor. To find the total magnetic field B created at some point by a current of finite size, we must sum up contributions from all current elements I dl that make up the current. That is, we must evaluate B by integrating Equation (--): ⃗ ⃗⃗⃗ ̂ ∫ where the integral is taken over the entire current distribution. This expression must be handled with special care because the integrand is a cross product and therefore a vector quantity. Although the Biot–Savart law was discussed for a current-carrying wire, it is also valid for a current consisting of charges flowing through space such as the particle beam in an accelerator. In that case, dl represents the length of a small segment of space in which the charges flow. Interesting similarities and differences exist between this Equation for the magnetic field due to a current element and Equation for the electric field due to a point charge ̂ The magnitude of the magnetic field varies as the inverse square of the distance from the source, as does the electric field due to a point charge. The directions of the two fields are quite different, however. The electric field created by a point charge is radial, but the magnetic field created by a current element is perpendicular to both the length element dl and the unit vector r as described by the cross product. Hence, if the conductor lies in the plane of the page as shown in Figure, dB points out of the page at P and into the page at P’.

Another difference between electric and magnetic fields is related to the source of the field. An electric field is established by an isolated electric charge. The Biot– Savart law gives the magnetic field of an isolated current element at some point, but such an isolated current element cannot exist the way an isolated electric charge can. A current element must be part of an extended current distribution because a complete circuit is needed for charges to flow. Therefore, the Biot– Savart law is only the first step in a calculation of a magnetic field; it must be followed by an integration over the current distribution as in Equation (--). Magnetic field surrounding thin, straight conductor Consider a thin, straight wire carrying a constant current I and placed along the x axis. The magnitude and direction of the magnetic field at point P due to this current is determined as follows. Fig (--)

From the Biot–Savart law, we expect that the magnitude of the field is proportional to the current in the wire and decreases as the distance a from the wire to point P increases. We are asked to find the magnetic field due to a simple current distribution, so this example is a typical problem for which the Biot–Savart law is appropriate. We must find the field contribution from a small element of current and then integrate over the current distribution. Let’s start by considering a length element dl located a distance r from P. The direction of the magnetic field at point P due to the current in this element is out of the page because dl x r^ is out of the page. In fact, because all the current elements I dl lie in the plane of the page, they all produce a magnetic field directed out of the page at point P. Therefore, the direction of the magnetic field at point P is out of the page and we need only find the magnitude of the field. We place the origin at O and let point P be along the positive y axis, with k being a unit vector pointing out of the page. Evaluate the cross product in the Biot–Savart law: ⃗⃗⃗ ̂ |⃗⃗⃗ ̂| ̂ * ( )+ ̂ ̂ Substitute into Equation (--) ̂ (1) ⃗⃗⃗⃗⃗ ̂ From the geometry in Figure (--), express r in terms of : (2) Notice that

() (3) This Equation shows that the magnitude of the magnetic field is proportional to the current and decreases with increasing distance from the wire, as expected. Equation 30.5 has the same mathematical form as the expression for the magnitude of the electric field due to a long charged wire Ampere’s law Figure (--)is a perspective view of the magnetic field surrounding a long, straight, current-carrying wire. Because of the wire’s symmetry, the magnetic field lines are circles concentric with the wire and lie in planes perpendicular to the wire. The magnitude of B is constant on any circle of radius a and is given by

A convenient rule for determining the direction of B is to grasp the wire with the right hand, positioning the thumb along the direction of the current. The four fingers wrap in the direction of the magnetic field. Figure (--) also shows that the magnetic field line has no beginning and no end. Rather, it forms a closed loop (closed path encloses open surface). That is a major difference between magnetic field lines and electric field lines, which begin on positive charges and end on negative charges. Oersted’s 1819 discovery about deflected compass needles demonstrates that a current-carrying conductor produces a magnetic field. The magnetic field produced by the current in the wire is consistent with the right-hand rule. When the current is reversed, the needles also reverse. Because the compass needles point in the direction of B, we conclude that the lines of B form circles around the wire. By symmetry, the magnitude of B is the same everywhere on a circular path centered on the wire and lying in a plane perpendicular to the wire. By varying the current and distance from the wire, we find that B is proportional to the current and inversely proportional to the distance from the wire. Now let’s evaluate the scalar product B.dL for a small length element dL on the circular closed path defined by the compass needles and sum the products for all elements over the closed circular path. Along this path, the vectors; dL (displacement vector through a closed path) and B (magnetic field vector as vector field) are parallel at each point, so B ⃗ ⃗⃗⃗⃗ Furthermore, the magnitude of B is constant on this circle and is given by Equation (--). Therefore, the sum of the products B ds over the closed path, which is equivalent to the line integral of B is ∮∮ Where the closed integral is over the circumference of the circular path of radius r. Although this result was calculated for the special case of a circular path surrounding a wire, it holds for a closed path of any shape (an amperian loop) surrounding a current that exists in an unbroken circuit.

The general case, known as Ampère’s law, can be stated as follows: The line integral of B.dL around any closed path equals oI, where I is the total steady current passing through any surface bounded by the closed path: ∮ Ampère’s law describes the creation of magnetic fields by all continuous current configurations, but at our mathematical level it is useful only for calculating the magnetic field of current configurations having a high degree of symmetry. Its use is similar to that of Gauss’s law in calculating electric fields for highly symmetric charge distributions. Displacement current and general form of Ampere’s law Previously, the Ampere's law to analyze the magnetic fields created by currents was discussed. It Is given by ∮ In this equation, the line integral is over any closed path through which conduction current passes, where conduction current is defined by the expression In this section, we use the term conduction current to refer to the current carried by charge carriers in the wire to distinguish it from a new type of current we shall introduce shortly.) We now show that Ampere’s law in this form is valid only if any electric fields present are constant in time. James Clerk Maxwell recognized this limitation and modified Ampere’s law to include time-varying electric fields. Consider a capacitor being charged as illustrated in following Figure.

When a conduction current is present, the charge on the positive plate changes, but no conduction current exists in the gap between the plates because there are no charge carriers in the gap. Now consider the two surfaces S1 and S2 in the Figure, bounded by the same path P. Ampere’s law states that ∮ Where I is the total current through any surface bounded by the path P. When the path P is considered to be the boundary of S1, the equation (-) is satisfy because the conduction current I passes through S1. When the path is considered to be the boundary of S2, then ∮ That is means the Ampere’s law did not satisfy because no conduction current passes through S2. Therefore, we have a contradictory situation that arises from the discontinuity of the current.

Maxwell solved this problem by postulating an additional term on the right side of Ampere’s law, which includes a factor called the displacement current Id defined as where o is the permittivity of free space and E is the electric flux through the surface bounded by the path of integration. As the capacitor is being charged (or discharged), the changing electric field between the plates may be considered equivalent to a current that acts as a continuation of the conduction current in the wire. When the expression for the displacement current given by Equation (-) is added to the conduction current on the right side of Ampere’s law, the difficulty represented in Figure (-) is resolved. No matter which surface bounded by the path P is chosen, either a conduction current or a displacement current passes through it. With this new term Id, we can express the general form of Ampere’s law (sometimes called the Ampere–Maxwell law) as ∮ () ∮ () ∮ We can understand the meaning of this expression by referring to the following Figure.

The electric flux through open surface S encloses with closed path L is ∫ Where S is the area of the capacitor plates and E is the magnitude of the uniform electric field between the plates. q is the charge on the plates at any instant. The displacement current through S is () That is, the displacement current Id through S is precisely equal to the conduction current I in the wires connected to the capacitor! By considering surface S, we can identify the displacement current as the source of the magnetic field on the surface boundary. The displacement current has its physical origin in the time-varying electric field. The central point of this formalism is that magnetic fields are produced both by conduction currents and by

time-varying electric fields. This result was a remarkable example of theoretical work by Maxwell, and it contributed to major advances in the understanding of electromagnetism.

Electromagnetics Faraday’s law of induction So far, our studies in electricity and magnetism have focused on the electric fields produced by stationary charges and the magnetic fields produced by moving charges. This chapter explores the effects produced by magnetic fields that vary in time. Experiments conducted by Michael Faraday in England in 1831 and independently by Joseph Henry in the United States that same year showed that an emf can be induced in a circuit by a changing magnetic field. The results of these experiments led to a very basic and important law of electromagnetism known as Faraday’s law of induction. An emf (and therefore a current as well) can be induced in various processes that involve a change in a magnetic flux. Electromotive force (emf)

Faraday’s law of induction To see how an electromotive force (emf) can be induced by a changing magnetic field, consider the experimental results obtained when a loop of wire is connected to a sensitive ammeter as illustrated in Active Figure (--). When a magnet is moved toward the loop, the reading on the ammeter changes from zero in one direction, arbitrarily shown as negative. When the magnet is brought to rest and held stationary relative to the loop, a reading of zero is observed. When the magnet is moved away from the loop, the reading on the ammeter changes in the opposite direction. Finally, when the magnet is held stationary and the loop is moved either toward or away from it, the reading changes from zero. From these observations, we conclude that the loop detects that the magnet is moving relative to it and we relate this detection to a change in magnetic field. Therefore, it seems that a relationship exists between current and changing magnetic field. These results are quite remarkable because a current is set up even though no batteries are present in the circuit! We call such a current an induced current and say that it is produced by an induced emf.

Now let’s describe an experiment conducted by Faraday and illustrated in Figure (- -). A primary coil is wrapped around an iron ring and connected to a switch and a battery. A current in the coil produces a magnetic field when the switch is closed. A secondary coil also is wrapped around the ring and is connected to a sensitive ammeter. No battery is present in the secondary circuit, and the secondary coil is not electrically connected to the primary coil. Any current detected in the secondary circuit must be induced by some external agent. Initially, you might guess that no current is ever detected in the secondary circuit. Something quite amazing happens when the switch in the primary circuit is either opened or thrown closed, however. At the instant the switch is closed, the ammeter reading changes from zero in one direction and then returns to zero. At the instant the switch is opened, the ammeter changes in the opposite direction and again returns to zero. Finally, the ammeter reads zero when there is either a steady current or no current in the primary circuit. To understand what happens in this experiment, note that when the switch is closed, the current in the primary circuit produces a magnetic field that penetrates the secondary circuit. Furthermore, when

the switch is thrown closed, the magnetic field produced by the current in the primary circuit changes from zero to some value over some finite time, and this changing field induces a current in the secondary circuit. As a result of these observations, Faraday concluded that an electric current can be induced in a loop by a changing magnetic field. The induced current exists only while the magnetic field through the loop is changing. Once the magnetic field reaches a steady value, the current in the loop disappears. In effect, the loop behaves as though a source of emf were connected to it for a short time. It is customary to say that an induced emf is produced in the loop by the changing magnetic field. The experiments shown have one thing in common: in each case, an emf is induced in a loop when the magnetic flux through the loop changes with time. In general, this emf is directly proportional to the time rate of change of the magnetic flux through the loop. This statement can be written mathematically as Faraday’s law of induction: If a coil consists of N loops with the same area and FB is the magnetic flux through one loop, an emf is induced in every loop. The loops are in series, so their emfs add; therefore, the total induced emf in the coil is given by The negative sign in Faraday’s law has a physical significance which would discuss in the nest section.

Suppose a loop (closed path L) enclosing an area (open surface S) lies in a uniform magnetic field B as in Figure (--). The magnetic flux B through the open surface S is given by Hence, the induced emf can be expressed as ) ( From this expression, we see that an emf can be induced in the circuit in several ways: 1- The magnitude of B can change with time. 2- The area enclosed by the loop can change with time. 3- The angle u between B and the normal to the loop can change with time. 4- Any combination of the above can occur.

Lenz’s law The negative sign in Equations is of important physical significance and will be discussed as follows. Faraday’s law indicates that the induced emf and the change in flux have opposite algebraic signs. This feature has a very real physical interpretation that has come to be known as Lenz’s law: The induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the open surface dS enclosed by the loop (closed path) L. That is, the induced current tends to keep the original magnetic flux through the loop from changing. We shall show that this law is a consequence of the law of conservation of energy. To understand Lenz’s law, suppose a conducting bar moving to the right on two parallel rails in the presence of a uniform magnetic field (the external magnetic field; Fig. (--.). As the bar moves to the right, the magnetic flux through the area enclosed by the circuit increases with time because the area increases.

Lenz’s law states that the induced current must be directed so that the magnetic field it produces opposes the change in the external magnetic flux. Because the magnetic flux due to an external field directed into the page is increasing, the induced current—if it is to oppose this change—must produce a field directed out of the page. Hence, the induced current must be directed counterclockwise when the bar moves to the right. (Use the right-hand rule to verify this direction.) If the bar is moving to the left as in Figure (--), the external magnetic flux through the area enclosed by the loop decreases with time. Because the field is directed into the page, the direction of the induced current must be clockwise if it is to produce a field that also is directed into the page. In either case, the induced current attempts to maintain the original flux through the area enclosed by the current loop. Let’s examine this situation using energy considerations. Suppose the bar is given a slight push to the right. In the preceding analysis, we found that this motion sets up a counterclockwise current in the loop. What happens if we assume the current is clockwise such that the direction of the magnetic force exerted on the bar is to the right? This force would accelerate the rod and increase its velocity, which in turn would cause the area enclosed by the loop to increase more rapidly. The result would be an increase in the induced current, which would cause an increase in the force, which would produce an increase in the current, and so on. In effect, the system would acquire energy with no input of energy. This behavior is clearly inconsistent with all experience and violates the law of conservation of energy. Therefore, the current must be counterclockwise.

Induced electric field We have seen that a changing magnetic flux induces an emf and a current in a conducting loop. In our study of electricity, we related a current to an electric field that applies electric forces on charged particles. In the same way, we can relate an induced current in a conducting loop to an electric field by claiming that an electric field is created in the conductor as a result of the changing magnetic flux. We also noted in our study of electricity that the existence of an electric field is independent of the presence of any test charges. This independence suggests that even in the absence of a conducting loop, a changing magnetic field generates an electric field in empty space. This induced electric field is nonconservative, unlike the electrostatic field produced by stationary charges. To illustrate this point, consider a conducting loop of radius r situated in a uniform magnetic field that is perpendicular to the plane of the loop as in Figure (--). If the magnetic field changes with time, an emf would create in the loop according to Faraday’s law, induced in the loop. The induction of a current in the loop implies the presence of an induced electric field E, which must be tangent to the loop because that is the direction in which the charges in the wire move in response to the electric force.

The work done by the electric field in moving a test charge q once around the loop is equal to q. Because the electric force acting on the charge is qE, the work done by the electric field in moving the charge once around the loop is given by ∮ ∮ () Where 2r is the circumference of the loop (circle). Using the relation between electric potential V (emf) and electric field E Then ∮ ∮ ( ) ( )( ) Then () Then Using Faraday’s law and the magnetic flux through open surface S encloses by the closed path L: And ∫ ∫ ()

Then the induced electric field is given by ( ( )) This equation gives the induced electric field created in the closed path according to Faraday’s law, it is differ from electrostatic field. If the time variation of the magnetic field is specified, the induced electric field can be calculated from this Equation. On the other hand, the induced emf for any closed path is the electric potential through the closed path. For a closed path the induced emf (electric potential) is related to the induced electric field as Where L is a closed path encloses an open surface S, then ∮ Substituting in Faraday’s law, then ∮ It is noted from this equation that the induced electric field E in this Equation is a nonconservative field that is generated by a changing magnetic field. The field E that satisfies this Equation cannot possibly be an electrostatic field because were the field electrostatic and hence conservative, the line integral of E over a closed loop would be zero, which would be in contradiction to this Equation. In more general cases, E may not be constant and the closed path may not be a circle. Hence, Faraday’s law of induction can be written in the general form as in the last equation.

Energy in a magnetic field In Chapter (--), we saw that an emf and a current are induced in a loop of wire when the magnetic flux through the area enclosed by the loop changes with time. This phenomenon of electromagnetic induction has some practical consequences. In this chapter, we first describe an effect known as self-induction, in which a time- varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. Self-induction is the basis of the inductor, an electrical circuit element. We discuss the energy stored in the magnetic field of an inductor and the energy density associated with the magnetic field. Self-induction and inductance we need to distinguish carefully between emfs and currents that are caused by physical sources such as batteries and those that are induced by changing magnetic fields. When we use a term (such as emf or current) without an adjective, we are describing the parameters associated with a physical source. We use the adjective induced to describe those emfs and currents caused by a changing magnetic field. Consider a circuit consisting of a switch, a resistor, and a source of emf as shown in Figure (--).

The circuit diagram is represented in perspective to show the orientations of some of the magnetic field lines due to the current in the circuit. When the switch is thrown to its closed position, the current does not immediately jump from zero to its maximum value /R. Faraday’s law of electromagnetic induction can be used to describe this effect as follows. As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time. This increasing flux creates an induced emf in the circuit. The direction of the induced emf is such that it would cause an induced current in the loop (if the loop did not already carry a current), which would establish a magnetic field opposing the change in the original magnetic field. Therefore, the direction of the induced emf is opposite the direction of the emf of the battery, which results in a gradual rather than instantaneous increase in the current to its final equilibrium value. Because of the direction of the induced emf, it is also called a back emf, similar to that in a motor. This effect is called self-induction because the changing flux through the circuit and the resultant induced emf arise from the circuit itself. The emf eL set up in this case is called a self-induced emf. To obtain a quantitative description of self-induction, recall from Faraday’s law that the induced emf is equal to the negative of the time rate of change of the magnetic flux. The magnetic flux is proportional to the magnetic field, which in turn is proportional to the current in the circuit. Therefore, a self-induced emf is always proportional to the time rate of change of the current. For any loop of wire, we can write this proportionality as Then where L is a proportionality constant, called the inductance of the loop, that depends on the geometry of the loop and other physical characteristics. If we consider a closely spaced coil of N turns (a toroid or an ideal solenoid) carrying a current I and containing N turns, Faraday’s law tells us that

Combining this expression with Equation (--) gives Then Recall that resistance is a measure of the opposition to current in comparison, Equation It shows us that inductance is a measure of the opposition to a change in current. The SI unit of inductance is the henry (H), which as we can see from Equation (--) is 1 volt-second per ampere: 1 H =1 V, s/A. The inductance of a coil depends on its geometry. This dependence is analogous to the capacitance of a capacitor depending on the geometry of its plates.

Energy in a magnetic field RL circuit If a circuit contains a coil such as a solenoid, the inductance of the coil prevents the current in the circuit from increasing or decreasing instantaneously. A circuit element that has a large inductance is called an inductor. We always assume the inductance of the remainder of a circuit is negligible compared with that of the inductor. Keep in mind, however, that even a circuit without a coil has some inductance that can affect the circuit’s behavior. Because the inductance of an inductor results in a back emf, an inductor in a circuit opposes changes in the current in that circuit. The inductor attempts to keep the current the same as it was before the change occurred. If the battery voltage in the circuit is increased so that the current rises, the inductor opposes this change and the rise is not instantaneous. If the battery voltage is decreased, the inductor causes a slow drop in the current rather than an immediate drop. Therefore, the inductor causes the circuit to be “sluggish” as it reacts to changes in the voltage. Consider the circuit shown in Active Figure (--).

It contains a battery of negligible internal resistance. This circuit is an RL circuit because the elements connected to the battery are a resistor and an inductor. The curved lines on switch S2 suggest this switch can never be open; it is always set to either a or b. (If the switch is connected to neither a nor b, any current in the circuit suddenly stops.) Suppose S2 is set to a and switch S1 is open for t < 0 and then thrown closed at t = 0. The current in the circuit begins to increase, and a back emf that opposes the increasing current is induced in the inductor. A battery in a circuit containing an inductor must provide more energy than in a circuit without the inductor. Part of the energy supplied by the battery appears as internal energy in the resistance in the circuit, and the remaining energy is stored in the magnetic field of the inductor. With this point in mind, let’s apply Kirchhoff’s loop rule to this circuit, traversing the circuit in the clockwise direction: Multiplying each term by I and rearranging the expression gives the electric power Recognizing I as the rate at which energy is supplied by the battery and I2R as the rate at which energy is delivered to the resistor, we see that LI(dI/dt) must represent the rate at which energy is being stored in the inductor. If U is the energy stored in the inductor at any time, we can write the rate dUB/dt at which energy is stored as To find the total energy stored in the inductor at any instant, let’s rewrite this expression as

∫ ∫ () Then This Equation represents the energy stored in the magnetic field of the inductor when the current is I. It is similar in form to Equation for the energy stored in the electric field of a capacitor. () In either case, energy is required to establish a field. Previously, it was derived the expressions for induction L of solenoid and the magnetic field B of solenoid as () And Then By substituting in equation of the energy stored in the magnetic field of the inductor, it gives

( ( ) )( ) The magnetic energy density uB is the energy stored in the magnetic field per unit volume V=Al of the solenoid ( ) ( )( ) ( ) Although this expression was derived for the special case of a solenoid, it is valid for any region of space in which a magnetic field exists. This Equation is similar in form to Equation for the energy per unit volume stored in an electric field, In both cases, the energy density is proportional to the square of the field magnitude. The magnetic field of a solenoid A solenoid is a long wire wound in the form of a helix. With this configuration, a reasonably uniform magnetic field can be produced in the space surrounded by the

turns of wire, which we shall call the interior of the solenoid, when the solenoid carries a current. When the turns are closely spaced, each can be approximated as a circular loop; the net magnetic field is the vector sum of the fields resulting from all the turns. Figure 30.16 shows the magnetic field lines surrounding a loosely wound solenoid. The field lines in the interior are nearly parallel to one another, are uniformly distributed, and are close together, indicating that the field in this space is strong and almost uniform. If the turns are closely spaced and the solenoid is of finite length, the magnetic field lines are as shown in Figure 30.17a. This field line distribution is similar to that surrounding a bar magnet (Fig. 30.17b). Hence, one end of the solenoid behaves like the north pole of a magnet and the opposite end behaves like the south pole.

As the length of the solenoid increases, the interior field becomes more uniform and the exterior field becomes weaker. An ideal solenoid is approached when the turns are closely spaced and the length is much greater than the radius of the turns. Figure 30.18 shows a longitudinal cross section of part of such a solenoid carrying a current I. In this case, the external field is close to zero and the interior field is uniform over a great volume.

Consider the amperian loop (loop 1) perpendicular to the page in Figure 30.18, surrounding the ideal solenoid. This loop encloses a small current as the charges in the wire move coil by coil along the length of the solenoid. Therefore, there is a nonzero magnetic field outside the solenoid. It is a weak field, with circular field lines, like those due to a line of current as in Figure 30.9. For an ideal solenoid, this weak field is the only field external to the solenoid. We could eliminate this field in Figure 30.18 by adding a second layer of turns of wire outside the first layer, with the current carried along the axis of the solenoid in the opposite direction compared with the first layer. Then the net current along the axis is zero.

We can use Ampère’s law to obtain a quantitative expression for the interior magnetic field in an ideal solenoid. Because the solenoid is ideal, B in the interior space is uniform and parallel to the axis and the magnetic field lines in the exterior space form circles around the solenoid. The planes of these circles are perpendicular to the page. Consider the rectangular path (loop 2) of length , and width w shown in Figure 30.18. Let’s apply Ampère’s law to this path by evaluating the integral of B ? dsS over each side of the rectangle. The contribution along side 3 is zero because the magnetic field lines are perpendicular to the path in this region. The contributions from sides 2 and 4 are both zero, again because B Sis perpendicular to dsS along these paths, both inside and outside the solenoid. Side 1 gives a contribution to the integral because along this path B is uniform and parallel to dsS. The integral over the closed rectangular path is therefore ∮∮ The right side of Ampère’s law involves the total current I through the area bounded by the path of integration. In this case, the total current through the rectangular path equals the current through each turn multiplied by the number of turns. If N is the number of turns in the length l, the total current through the rectangle is NI. Therefore, Ampère’s law applied to this path gives ∮ () Equating the two equations, then () where n= N/l, is the number of turns per unit length.

Inductance of solenoid In this section it could be determine the inductance of solenoid as follows. Consider a uniformly wound solenoid having N turns and length l. Assume the length is much longer than the radius of the windings and the core of the solenoid is air, as shown in Fig. (--). The magnetic field lines from each turn of the solenoid pass through all the turns, so an induced emf in each coil opposes changes in the current. The magnetic flux through each turn of area A in the solenoid is given by () Then, the inductance is ( () ) ()

Maxwell’s equations and electromagnetic waves The mechanical waves by definition, are the propagation of mechanical disturbances, such as sound waves, water waves, and waves on a string, requires the presence of a medium. This chapter is concerned with the properties of electromagnetic waves, which (unlike mechanical waves) can propagate through empty space. Maxwell’s equations form the theoretical basis of all electromagnetic phenomena. These equations predict the existence of electromagnetic waves that propagate through space at the speed of light c. Heinrich Hertz confirmed Maxwell’s prediction when he generated and detected electromagnetic waves in 1887. That discovery has led to many practical communication systems, including radio, television, cell phone systems, wireless Internet connectivity, and optoelectronics. Next, we learn how electromagnetic waves are generated by oscillating electric charges. The waves radiated from the oscillating charges can be detected at great distances. Furthermore, because electromagnetic waves carry energy and momentum, they can exert pressure on a surface. The chapter concludes with a look at many frequencies covered by electromagnetic waves. Maxwell’s equations in symmetrical form in vacuum We now present four equations that are regarded as the basis of all electrical and magnetic phenomena. These equations, developed by Maxwell, are as fundamental to electromagnetic phenomena as Newton’s laws are to mechanical phenomena. In fact, the theory that Maxwell developed was more far-reaching than even he imagined because it turned out to be in agreement with the special theory of relativity, as Einstein showed in 1905. Maxwell’s equations represent the laws of electricity and magnetism, but they have additional important consequences. For simplicity, we present Maxwell’s equations as applied to free space, that is, in the absence of any dielectric or magnetic material. The four equations are

Gauss’s law of electrostatics: for close surface A that enclose volume V (A(V)), the total electric flux through any closed surface A equals the net charge inside that surface divided by o. This law relates an electric field to the charge distribution that creates it. ∮ Gauss’s law of magnetostatics: for closed surface A that encloses a volume V (A(V)), the net magnetic flux through a closed surface is zero. That is, the number of magnetic field lines that enter a closed volume must equal the number that leave that volume, which implies that magnetic field lines cannot begin or end at any point. If they did, it would mean that isolated magnetic monopoles existed at those points. That isolated magnetic monopoles have not been observed in nature. ∮ Faraday’s law of induction: for enclosed path L encloses open surface S (L(S)). It describes the creation of an electric field by a changing magnetic flux. This law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of magnetic flux through any surface bounded by that path. One consequence of Faraday’s law is the current induced in a conducting loop placed in a time-varying magnetic field. ∮ Ampere-Maxwell law of electrodynamics: for enclosed path L encloses open surface S (L(S)). It describes the creation of a magnetic field by a changing electric field and by electric current: the line integral of the magnetic field around any closed path is the sum of o multiplied by the net current through that path and oo multiplied by the rate of change of electric flux through any surface bounded by that path.

∮ Lorentz force law: Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated from the expression Where Fe and Fm are the electric and magnetic forces. Maxwell’s equations, together with this force law, completely describe all classical electromagnetic interactions in a vacuum. Notice the symmetry of Maxwell’s equations. Equations 34.4 and 34.5 are symmetric, apart from the absence of the term for magnetic monopoles in Equation 34.5. Furthermore, Equations 34.6 and 34.7 are symmetric in that the line integrals of E and B around a closed path are related to the rate of change of magnetic flux and electric flux, respectively. Maxwell’s equations are of fundamental importance not only to electromagnetism, but to all science. Hertz once wrote, “One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we put into them.” In the next section, we show that Equations 34.6 and 34.7 can be combined to obtain a wave equation for both the electric field and the magnetic field. In empty space, where q = 0 and I = 0, the solution to these two equations shows that the speed at which electromagnetic waves travel equals the measured speed of light. This result led Maxwell to predict that light waves are a form of electromagnetic radiation.

Solution of Maxwell’s equations Plane electromagnetic waves The properties of electromagnetic waves can be deduced from Maxwell’s equations. One approach to deriving these properties is to solve the second- order differential equation obtained from Maxwell’s third and fourth equations. A rigorous mathematical treatment of that sort is beyond the scope of this text. To circumvent this problem, let’s assume the vectors for the electric field and magnetic field in an electromagnetic wave have a specific space–time behavior that is simple but consistent with Maxwell equations. To understand the prediction of electromagnetic waves more fully, let’s focus our attention on an electromagnetic wave that travels in the x direction (the direction of propagation). For this wave, the electric field E is in the y direction and the magnetic field B is in the z direction as shown in Active Figure 34.5. Such waves, in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes, are said to be linearly polarized waves. Furthermore, let’s assume the field magnitudes E and B depend on x and t only, not on the y or z coordinate. Let’s also imagine that the source of the electromagnetic waves is such that a wave radiated from any position in the yz plane (not only from the origin as might be suggested by Active Fig. 34.5) propagates in the x direction and all such waves are emitted in phase. If we define a ray as the line along which the wave travels, all rays for these waves are parallel. This entire collection of waves is often called a plane wave. A surface connecting points of equal