13.1 Faraday’s Law

The first productive experiments concerning the effects of time-varying magnetic fields were performed by Michael Faraday in 1831. One of his early experiments is represented in Figure 13.2. An emf is induced when the magnetic field in the coil is changed by pushing a bar magnet into or out of the coil. Emfs of opposite signs are produced by motion in opposite directions, and the directions of emfs are also reversed by reversing poles. The same results are produced if the coil is moved rather than the magnet—it is the relative motion that is important. The faster the motion, the greater the emf, and there is no emf when the magnet is stationary relative to the coil.

Figure A shows the magnet with the North pole facing up moved into the coil. Figure B shows the magnet with the North pole facing up moved out of the coil. Figure C shows the magnet with the South pole facing up moved into the coil. Figure D shows the magnet with the South pole facing up moved out of the coil. Figure E shows the magnet with the North pole facing up hold stationary in the coil. — **Figure 13.2** Movement of a magnet relative to a coil produces emfs as shown (a–d). The same emfs are produced if the coil is moved relative to the magnet. This short-lived emf is only present during the motion. The greater the speed, the greater the magnitude of the emf, and the emf is zero when there is no motion, as shown in (e).

Faraday also discovered that a similar effect can be produced using two circuits—a changing current in one circuit induces a current in a second, nearby circuit. For example, when the switch is closed in circuit 1 of Figure 13.3(a), the ammeter needle of circuit 2 momentarily deflects, indicating that a short-lived current surge has been induced in that circuit. The ammeter needle quickly returns to its original position, where it remains. However, if the switch of circuit 1 is now suddenly opened, another short-lived current surge in the direction opposite from before is observed in circuit 2.

Figure A shows the circuit in which switch is getting closed causing current to flow clockwise through it. Figure B shows the circuit in which switch is kept closed. There is no current in the circuit. Figure C shows the circuit in which switch is getting opened causing current to flow counter clockwise through it. — **Figure 13.3** (a) Closing the switch of circuit 1 produces a short-lived current surge in circuit 2. (b) If the switch remains closed, no current is observed in circuit 2. (c) Opening the switch again produces a short-lived current in circuit 2 but in the opposite direction from before.

Faraday realized that in both experiments, a current flowed in the circuit containing the ammeter only when the magnetic field in the region occupied by that circuit was changing. As the magnet of the figure was moved, the strength of its magnetic field at the loop changed; and when the current in circuit 1 was turned on or off, the strength of its magnetic field at circuit 2 changed. Faraday was eventually able to interpret these and all other experiments involving magnetic fields that vary with time in terms of the following law:

The magnetic flux is a measurement of the amount of magnetic field lines through a given surface area, as seen in Figure 13.4. This definition is similar to the electric flux studied earlier. This means that if we have

$Φ_{m} = \int_{S} \vec{B} \cdot \hat{n} d A,$

(13.1)

then the induced emf or the voltage generated by a conductor or coil moving in a magnetic field is

$ε = - \frac{d}{d t} \int_{S} \vec{B} \cdot \hat{n} d A = - \frac{d Φ_{m}}{d t} .$

(13.2)

The negative sign describes the direction in which the induced emf drives current around a circuit. However, that direction is most easily determined with a rule known as Lenz’s law, which we will discuss shortly.

Figure shows a uniform magnetic field B cutting through a surface area A. — **Figure 13.4** The magnetic flux is the amount of magnetic field lines cutting through a surface area A defined by the unit area vector $n ^.$ If the angle between the unit area $n ^$ and magnetic field vector $\vec{B}$ are parallel or antiparallel, as shown in the diagram, the magnetic flux is the highest possible value given the values of area and magnetic field.

Part (a) of Figure 13.5 depicts a circuit and an arbitrary surface S that it bounds. Notice that S is an open surface. It can be shown that any open surface bounded by the circuit in question can be used to evaluate $Φ_{m} .$ For example, $Φ_{m}$ is the same for the various surfaces $S sub 1, S sub 2, …$ of part (b) of the figure.

Figure A shows the circuit bounding an arbitrary open surface S. The planar area bounded by the circuit is not part of S. Figure B shows three arbitrary open surfaces S1, S2, and S3 bounded by the same circuit. — **Figure 13.5** (a) A circuit bounding an arbitrary open surface S. The planar area bounded by the circuit is not part of S. (b) Three arbitrary open surfaces bounded by the same circuit. The value of $Φ_{m}$ is the same for all these surfaces.

The SI unit for magnetic flux is the weber (Wb),

$1 Wb = 1 T \cdot m^{2} .$

Occasionally, the magnetic field unit is expressed as webers per square meter ( $Wb/m squared$ ) instead of teslas, based on this definition. In many practical applications, the circuit of interest consists of a number N of tightly wound turns (see Figure 13.6). Each turn experiences the same magnetic flux. Therefore, the net magnetic flux through the circuits is N times the flux through one turn, and Faraday’s law is written as

$ε = - \frac{d}{d t} (N Φ_{m}) = − N \frac{d Φ_{m}}{d t} .$

Example 1 A Square Coil in a Changing Magnetic Field

The square coil of Figure 13.6 has sides $l equals 0.25 m$ long and is tightly wound with $N equals 200$ turns of wire. The resistance of the coil is $R equals 5.0 Ω.$ The coil is placed in a spatially uniform magnetic field that is directed perpendicular to the face of the coil and whose magnitude is decreasing at a rate $d B / d t equals −0.040 T / s.$ (a) What is the magnitude of the emf induced in the coil? (b) What is the magnitude of the current circulating through the coil?

Figure 13.6 A square coil with N turns of wire with uniform magnetic field

\vec{B}

directed in the downward direction, perpendicular to the coil.

Strategy The area vector, or $n ^$ direction, is perpendicular to area covering the loop. We will choose this to be pointing downward so that $\vec{B}$ is parallel to $n ^$ and that the flux turns into multiplication of magnetic field times area. The area of the loop is not changing in time, so it can be factored out of the time derivative, leaving the magnetic field as the only quantity varying in time. Lastly, we can apply Ohm’s law once we know the induced emf to find the current in the loop.

Solution

The flux through one turn is
$Φ_{m} = B A = B l^{2},$
so we can calculate the magnitude of the emf from Faraday’s law. The sign of the emf will be discussed in the next section, on Lenz’s law:
$\begin{matrix} | ε | & = | − N \frac{d Φ_{m}}{d t} | = N l^{2} \frac{d B}{d t} \\ = (200) {(0.25 m)}^{2} (0.040 T/s) = 0.50 V . \end{matrix}$
The magnitude of the current induced in the coil is
$I = \frac{ε}{R} = \frac{0.50 V}{5.0 Ω} = 0.10 A .$

Significance If the area of the loop were changing in time, we would not be able to pull it out of the time derivative. Since the loop is a closed path, the result of this current would be a small amount of heating of the wires until the magnetic field stops changing. This may increase the area of the loop slightly as the wires are heated.

Summary

The magnetic flux through an enclosed area is defined as the amount of field lines cutting through a surface area A defined by the unit area vector.
The units for magnetic flux are webers, where $1 Wb = 1 T \cdot m^{2} .$
The induced emf in a closed loop due to a change in magnetic flux through the loop is known as Faraday’s law. If there is no change in magnetic flux, no induced emf is created.

Conceptual Questions

A stationary coil is in a magnetic field that is changing with time. Does the emf induced in the coil depend on the actual values of the magnetic field?

The emf depends on the rate of change of the magnetic field.

In Faraday’s experiments, what would be the advantage of using coils with many turns?

A copper ring and a wooden ring of the same dimensions are placed in magnetic fields so that there is the same change in magnetic flux through them. Compare the induced electric fields and currents in the rings.

Both have the same induced electric fields; however, the copper ring has a much higher induced emf because it conducts electricity better than the wooden ring.

Discuss the factors determining the induced emf in a closed loop of wire.

(a) Does the induced emf in a circuit depend on the resistance of the circuit? (b) Does the induced current depend on the resistance of the circuit?

a. no; b. yes

How would changing the radius of loop D shown below affect its emf, assuming C and D are much closer together compared to their radii?

Figure shows a circuit that consists of a resistor, capacitor, opened switch and a loop C. A loop D is located next to the loop C.

Can there be an induced emf in a circuit at an instant when the magnetic flux through the circuit is zero?

As long as the magnetic flux is changing from positive to negative or negative to positive, there could be an induced emf.

Does the induced emf always act to decrease the magnetic flux through a circuit?

How would you position a flat loop of wire in a changing magnetic field so that there is no induced emf in the loop?

Position the loop so that the field lines run perpendicular to the area vector or parallel to the surface.

The normal to the plane of a single-turn conducting loop is directed at an angle $θ$ to a spatially uniform magnetic field $\vec{B} .$ It has a fixed area and orientation relative to the magnetic field. Show that the emf induced in the loop is given by $ε equals open parenthesis d B / d t close parenthesis open parenthesis A cos θ close parenthesis,$ where A is the area of the loop.

Problems

A 50-turn coil has a diameter of 15 cm. The coil is placed in a spatially uniform magnetic field of magnitude 0.50 T so that the face of the coil and the magnetic field are perpendicular. Find the magnitude of the emf induced in the coil if the magnetic field is reduced to zero uniformly in (a) 0.10 s, (b) 1.0 s, and (c) 60 s.

Repeat your calculations of the preceding problem’s time of 0.1 s with the plane of the coil making an angle of (a) $30 °,$ (b) $60 °,$ and (c) $90 °$ with the magnetic field.

a. 3.8 V; b. 2.2 V; c. 0 V

A square loop whose sides are 6.0-cm long is made with copper wire of radius 1.0 mm. If a magnetic field perpendicular to the loop is changing at a rate of 5.0 mT/s, what is the current in the loop?

The magnetic field through a circular loop of radius 10.0 cm varies with time as shown below. The field is perpendicular to the loop. Plot the magnitude of the induced emf in the loop as a function of time.

Figure shows the magnetic field in milliTesla plotted as a function of time in ms. Magnetic field is zero when the time is equal to zero. It increases linearly with time reaching 3 milliTesla and 2 ms. It remains the same till 5 ms and then decreases linearly to 0 and 6 ms.

$\begin{matrix} B & = & 1.5 t, 0 \leq t < 2.0 ms, B = 3.0 mT, 2.0 ms \leq t \leq 5.0 ms, \\ B & = & −3.0 t + 18 mT, 5.0 ms < t \leq 6.0 ms, \\ ε & = & - \frac{d Φ_{m}}{d t} = - \frac{d (B A)}{d t} = − A \frac{d B}{d t}, \\ ε & = & − π {(0.100 m)}^{2} (1.5 T/s) \\ = & −47 mV (0 \leq t < 2.0 ms), \\ ε & = & π {(0.100 m)}^{2} (0) = 0 (2.0 ms \leq t \leq 5.0 ms), \\ ε & = & − π {(0.100 m)}^{2} (−3.0 T/s) = 94 mV (5.0 ms < t < 6.0 ms) . \end{matrix}$

Figure shows the Emf in mV plotted as a function of time in ms. Emf is equal to -47 mV when the time is equal to zero. It increases in a step fashion to 0 when the time reaches 2 ms. Emf remains the same till 5 ms and then increases in a step fashion to 94 mV. It stays constant till time reaches 6 ms.

The accompanying figure shows a single-turn rectangular coil that has a resistance of $2.0 Ω.$ The magnetic field at all points inside the coil varies according to $B = B_{0} e^{− α t},$ where $B sub 0 equals 0.25 T$ and $α equals 200 Hz.$ What is the current induced in the coil at (a) $t equals 0.001 s$ , (b) 0.002 s, (c) 2.0 s?

Figure shows a square coil with the sides of 2.0 and 5.0 cm. A uniform magnetic field B is directed perpendicular to the coil.

How would the answers to the preceding problem change if the coil consisted of 20 closely spaced turns?

Each answer is 20 times the previously given answers.

A long solenoid with $n equals 10$ turns per centimeter has a cross-sectional area of $5.0 cm squared$ and carries a current of 0.25 A. A coil with five turns encircles the solenoid. When the current through the solenoid is turned off, it decreases to zero in 0.050 s. What is the average emf induced in the coil?

A rectangular wire loop with length a and width b lies in the xy-plane, as shown below. Within the loop there is a time-dependent magnetic field given by $\vec{B} (t) = C ((x cos ω t) \hat{i} + (y sin ω t) \hat{k})$ , with $\vec{B} (t)$ in tesla. Determine the emf induced in the loop as a function of time.

Figure shows a rectangular wire loop with length a and width b lies in the xy-plane.

$\begin{matrix} \hat{n} & = & \hat{k}, d Φ_{m} = C y sin (ω t) d x d y, \\ Φ_{m} & = & \frac{C a b^{2} sin (ω t)}{2}, \\ ε & = & - \frac{C a b^{2} ω cos (ω t)}{2} . \end{matrix}$

The magnetic field perpendicular to a single wire loop of diameter 10.0 cm decreases from 0.50 T to zero. The wire is made of copper and has a diameter of 2.0 mm. How much charge moves through the wire while the field is changing?