13.3 Motional Emf

Magnetic flux depends on three factors: the strength of the magnetic field, the area through which the field lines pass, and the orientation of the field with the surface area. If any of these quantities varies, a corresponding variation in magnetic flux occurs. So far, we’ve only considered flux changes due to a changing field. Now we look at another possibility: a changing area through which the field lines pass including a change in the orientation of the area.

Two examples of this type of flux change are represented in Figure 13.11. In part (a), the flux through the rectangular loop increases as it moves into the magnetic field, and in part (b), the flux through the rotating coil varies with the angle $θ$ .

Figure A shows a rectangular loop moving into a perpendicular magnetic field. Figure B shows a square loop rotating in a magnetic field. — **Figure 13.11** (a) Magnetic flux changes as a loop moves into a magnetic field; (b) magnetic flux changes as a loop rotates in a magnetic field.

Now let’s look at a conducting rod pulled in a circuit, changing magnetic flux. The area enclosed by the circuit ‘MNOP’ of Figure 13.12 is lx and is perpendicular to the magnetic field, so we can simplify the integration of Equation 13.1 into a multiplication of magnetic field and area. The magnetic flux through the open surface is therefore

$Φ_{m} = B l x .$

Since B and l are constant and the velocity of the rod is $v equals d x / d t,$ we can now restate Faraday’s law, Equation 13.2, for the magnitude of the emf in terms of the moving conducting rod as

$ε = \frac{d Φ_{m}}{d t} = B l \frac{d x}{d t} = B l v .$

The current induced in the circuit is the emf divided by the resistance or

$I = \frac{B l v}{R} .$

Furthermore, the direction of the induced emf satisfies Lenz’s law, as you can verify by inspection of the figure.

This calculation of motionally induced emf is not restricted to a rod moving on conducting rails. With $\vec{F} = q \vec{v} \times \vec{B}$ as the starting point, it can be shown that $ε equals − d Φ sub m / d t$ holds for any change in flux caused by the motion of a conductor. We saw in Faraday’s Law that the emf induced by a time-varying magnetic field obeys this same relationship, which is Faraday’s law. Thus Faraday’s law holds for all flux changes, whether they are produced by a changing magnetic field, by motion, or by a combination of the two.

Figure shows a conducting rod that is pushed to the right through the perpendicular magnetic field at constant velocity. The resulting change in the magnetic flux induces a current in the circuit. — **Figure 13.12** A conducting rod is pushed to the right at constant velocity. The resulting change in the magnetic flux induces a current in the circuit.

From an energy perspective, ${\vec{F}}_{a}$ produces power $F_{a} v,$ and the resistor dissipates power $I^{2} R$ . Since the rod is moving at constant velocity, the applied force $F_{a}$ must balance the magnetic force $F_{m} = I l B$ on the rod when it is carrying the induced current I. Thus the power produced is

$F_{a} v = I l B v = \frac{B l v}{R} \cdot l B v = \frac{l^{2} B^{2} v^{2}}{R} .$

The power dissipated is

$P = I^{2} R = {(\frac{B l v}{R})}^{2} R = \frac{l^{2} B^{2} v^{2}}{R} .$

In satisfying the principle of energy conservation, the produced and dissipated powers are equal.

This principle can be seen in the operation of a rail gun. A rail gun is an electromagnetic projectile launcher that uses an apparatus similar to Figure 13.12 and is shown in schematic form in Figure 13.13. The conducting rod is replaced with a projectile or weapon to be fired. So far, we’ve only heard about how motion causes an emf. In a rail gun, the optimal shutting off/ramping down of a magnetic field decreases the flux in between the rails, causing a current to flow in the rod (armature) that holds the projectile. This current through the armature experiences a magnetic force and is propelled forward. Rail guns, however, are not used widely in the military due to the high cost of production and high currents: Nearly one million amps is required to produce enough energy for a rail gun to be an effective weapon.

Figure shows a schematic drawing of the rail gun. An armature is placed between two rails of opposite charge. Magnetic field is generated by currents in rails and interacts with the current in armature, generating the force. — **Figure 13.13** Current through two rails drives a conductive projectile forward by the magnetic force created.

We can calculate a motionally induced emf with Faraday’s law even when an actual closed circuit is not present. We simply imagine an enclosed area whose boundary includes the moving conductor, calculate $Φ_{m}$ , and then find the emf from Faraday’s law. For example, we can let the moving rod of Figure 13.14 be one side of the imaginary rectangular area represented by the dashed lines. The area of the rectangle is lx, so the magnetic flux through it is $Φ_{m} = B l x .$ Differentiating this equation, we obtain

$\frac{d Φ_{m}}{d t} = B l \frac{d x}{d t} = B l v,$

which is identical to the potential difference between the ends of the rod that we determined earlier.

Figure shows a conducting rod of the length l that is pushed to the right for the distance x through the perpendicular magnetic field at constant velocity. — **Figure 13.14** With the imaginary rectangle shown, we can use Faraday’s law to calculate the induced emf in the moving rod.

Motional emfs in Earth’s weak magnetic field are not ordinarily very large, or we would notice voltage along metal rods, such as a screwdriver, during ordinary motions. For example, a simple calculation of the motional emf of a 1.0-m rod moving at 3.0 m/s perpendicular to the Earth’s field gives

$emf = B ℓ v = (5.0 \times 10^{−5} T) (1.0 m) (3.0 m/s) = 150 μ V .$

This small value is consistent with experience. There is a spectacular exception, however. In 1992 and 1996, attempts were made with the space shuttle to create large motional emfs. The tethered satellite was to be let out on a 20-km length of wire, as shown in Figure 13.15, to create a 5-kV emf by moving at orbital speed through Earth’s field. This emf could be used to convert some of the shuttle’s kinetic and potential energy into electrical energy if a complete circuit could be made. To complete the circuit, the stationary ionosphere was to supply a return path through which current could flow. (The ionosphere is the rarefied and partially ionized atmosphere at orbital altitudes. It conducts because of the ionization. The ionosphere serves the same function as the stationary rails and connecting resistor in Figure 13.13, without which there would not be a complete circuit.) Drag on the current in the cable due to the magnetic force $F = I ℓ B sin θ$ does the work that reduces the shuttle’s kinetic and potential energy, and allows it to be converted into electrical energy. Both tests were unsuccessful. In the first, the cable hung up and could only be extended a couple of hundred meters; in the second, the cable broke when almost fully extended. Example 1 indicates feasibility in principle.

Figure is a schematic drawing that shows tethered satellite that launched and returns to the space shuttle while moving through the Earth’s magnetic field. — **Figure 13.15** Motional emf as electrical power conversion for the space shuttle was the motivation for the tethered satellite experiment. A 5-kV emf was predicted to be induced in the 20-km tether while moving at orbital speed in Earth’s magnetic field. The circuit is completed by a return path through the stationary ionosphere.

Example 2 A Metal Rod Rotating in a Magnetic Field

Part (a) of Figure 13.16 shows a metal rod OS that is rotating in a horizontal plane around point O. The rod slides along a wire that forms a circular arc PST of radius r. The system is in a constant magnetic field $\vec{B}$ that is directed out of the page. (a) If you rotate the rod at a constant angular velocity $ω$ , what is the current I in the closed loop OPSO? Assume that the resistor R furnishes all of the resistance in the closed loop. (b) Calculate the work per unit time that you do while rotating the rod and show that it is equal to the power dissipated in the resistor.

Figure 13.16 (a) The end of a rotating metal rod slides along a circular wire in a horizontal plane. (b) The induced current in the rod. (c) The magnetic force on an infinitesimal current segment.

Strategy The magnetic flux is the magnetic field times the area of the quarter circle or $A equals r squared θ / 2.$ When finding the emf through Faraday’s law, all variables are constant in time but $θ$ , with $ω equals d θ / d t.$ To calculate the work per unit time, we know this is related to the torque times the angular velocity. The torque is calculated by knowing the force on a rod and integrating it over the length of the rod.

Solution

From geometry, the area of the loop OPSO is $A = \frac{r^{2} θ}{2} .$ Hence, the magnetic flux through the loop is
$Φ_{m} = B A = B \frac{r^{2} θ}{2} .$
Differentiating with respect to time and using $ω equals d θ / d t,$ we have
$ε = | \frac{d Φ_{m}}{d t} | = \frac{B r^{2} ω}{2} .$
When divided by the resistance R of the loop, this yields for the magnitude of the induced current
$I = \frac{ε}{R} = \frac{B r^{2} ω}{2 R} .$
As $θ$ increases, so does the flux through the loop due to $\vec{B} .$ To counteract this increase, the magnetic field due to the induced current must be directed into the page in the region enclosed by the loop. Therefore, as part (b) of Figure 13.16 illustrates, the current circulates clockwise.
You rotate the rod by exerting a torque on it. Since the rod rotates at constant angular velocity, this torque is equal and opposite to the torque exerted on the current in the rod by the original magnetic field. The magnetic force on the infinitesimal segment of length dx shown in part (c) of Figure 13.16 is $d F_{m} = I B d x,$ so the magnetic torque on this segment is
$d τ_{m} = x \cdot d F_{m} = I B x d x .$
The net magnetic torque on the rod is then
$τ_{m} = \int_{0}^{r} d τ_{m} = I B \int_{0}^{r} x d x = \frac{1}{2} I B r^{2} .$
The torque $τ$ that you exert on the rod is equal and opposite to $τ_{m},$ and the work that you do when the rod rotates through an angle $d θ$ is $d W = τ d θ .$ Hence, the work per unit time that you do on the rod is
$\frac{d W}{d t} = τ \frac{d θ}{d t} = \frac{1}{2} I B r^{2} \frac{d θ}{d t} = \frac{1}{2} (\frac{B r^{2} ω}{2 R}) B r^{2} ω = \frac{B^{2} r^{4} ω^{2}}{4 R},$
where we have substituted for I. The power dissipated in the resistor is $P = I^{2} R$ , which can be written as
$P = {(\frac{B r^{2} ω}{2 R})}^{2} R = \frac{B^{2} r^{4} ω^{2}}{4 R} .$
Therefore, we see that
$P = \frac{d W}{d t} .$
Hence, the power dissipated in the resistor is equal to the work per unit time done in rotating the rod.

Significance An alternative way of looking at the induced emf from Faraday’s law is to integrate in space instead of time. The solution, however, would be the same. The motional emf is

$| ε | equals ∫ B v d l.$

The velocity can be written as the angular velocity times the radius and the differential length written as dr. Therefore,

$| ε | = B \int v d r = B ω \int_{0}^{l} r d r = \frac{1}{2} B ω l^{2},$

which is the same solution as before.

Example 3 A Rectangular Coil Rotating in a Magnetic Field

A rectangular coil of area A and N turns is placed in a uniform magnetic field $B → equals B j ^,$ as shown in Figure 13.17. The coil is rotated about the z-axis through its center at a constant angular velocity $ω .$ Obtain an expression for the induced emf in the coil.

Figure 13.17 A rectangular coil rotating in a uniform magnetic field.

Strategy According to the diagram, the angle between the perpendicular to the surface ( $n ^$ ) and the magnetic field $(\vec{B})$ is $θ$ . The dot product of $B → times n ^$ simplifies to only the $cos θ$ component of the magnetic field, namely where the magnetic field projects onto the unit area vector $n ^$ . The magnitude of the magnetic field and the area of the loop are fixed over time, which makes the integration simplify quickly. The induced emf is written out using Faraday’s law.

Solution When the coil is in a position such that its normal vector $n ^$ makes an angle $θ$ with the magnetic field $\vec{B},$ the magnetic flux through a single turn of the coil is

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

From Faraday’s law, the emf induced in the coil is

$ε = − N \frac{d Φ_{m}}{d t} = N B A sin θ \frac{d θ}{d t} .$

The constant angular velocity is $ω equals d θ / d t.$ The angle $θ$ represents the time evolution of the angular velocity or $ω t$ . This is changes the function to time space rather than $θ$ . The induced emf therefore varies sinusoidally with time according to

$ε equals ε sub 0 sin ω t,$

where $ε sub 0 equals N B A ω.$

Significance If the magnetic field strength or area of the loop were also changing over time, these variables wouldn’t be able to be pulled out of the time derivative to simplify the solution as shown. This example is the basis for an electric generator, as we will give a full discussion in Electric Generators and Back Emf.

Summary

The relationship between an induced emf $ε$ in a wire moving at a constant speed v through a magnetic field B is given by $ε = B l v .$
An induced emf from Faraday’s law is created from a motional emf that opposes the change in flux.

Conceptual Questions

A bar magnet falls under the influence of gravity along the axis of a long copper tube. If air resistance is negligible, will there be a force to oppose the descent of the magnet? If so, will the magnet reach a terminal velocity?

Around the geographic North Pole (or magnetic South Pole), Earth’s magnetic field is almost vertical. If an airplane is flying northward in this region, which side of the wing is positively charged and which is negatively charged?

Positive charges on the wings would be to the west, or to the left of the pilot while negative charges would be pulled east or to the right of the pilot. Thus, the left hand tips of the wings would be positive and the right hand tips would be negative.

A wire loop moves translationally (no rotation) in a uniform magnetic field. Is there an emf induced in the loop?

Problems

An automobile with a radio antenna 1.0 m long travels at 100.0 km/h in a location where the Earth’s horizontal magnetic field is $5.5 \times 10^{− 5} T .$ What is the maximum possible emf induced in the antenna due to this motion?

0.0015 V

The rectangular loop of N turns shown below moves to the right with a constant velocity $\vec{v}$ while leaving the poles of a large electromagnet. (a) Assuming that the magnetic field is uniform between the pole faces and negligible elsewhere, determine the induced emf in the loop. (b) What is the source of work that produces this emf?

Figure shows the rectangular loop (short side has a length l, long side has a length a) of N turns that moves to the right with a constant velocity v while leaving the uniform magnetic field.

Suppose the magnetic field of the preceding problem oscillates with time according to $B equals B sub 0 sin ω t.$ What then is the emf induced in the loop when its trailing side is a distance d from the right edge of the magnetic field region?

$ε = − B_{0} l d ω cos (Ω t) ld + B_{0} sin (Ω t) lv$

A coil of 1000 turns encloses an area of $25 cm squared$ . It is rotated in 0.010 s from a position where its plane is perpendicular to Earth’s magnetic field to one where its plane is parallel to the field. If the strength of the field is $6.0 \times 10^{−5} T,$ what is the average emf induced in the coil?

In the circuit shown in the accompanying figure, the rod slides along the conducting rails at a constant velocity $\vec{v} .$ The velocity is in the same plane as the rails and directed at an angle $θ$ to them. A uniform magnetic field $\vec{B}$ is directed out of the page. What is the emf induced in the rod?

Figure shows the rod that slides along the conducting rails at a constant velocity v in a uniform perpendicular magnetic field. Distance between the rails is l. Angle between the direction of movement of the rod and the rails is theta.

$ε = B l v cos θ$

The rod shown in the accompanying figure is moving through a uniform magnetic field of strength $B equals 0.50 T$ with a constant velocity of magnitude $v equals 8.0 m / s.$ What is the potential difference between the ends of the rod? Which end of the rod is at a higher potential?

Figure shows the 5 cm long rod of that moves to the right at a constant velocity v in a uniform perpendicular magnetic field.

A 25-cm rod moves at 5.0 m/s in a plane perpendicular to a magnetic field of strength 0.25 T. The rod, velocity vector, and magnetic field vector are mutually perpendicular, as indicated in the accompanying figure. Calculate (a) the magnetic force on an electron in the rod, (b) the electric field in the rod, and (c) the potential difference between the ends of the rod. (d) What is the speed of the rod if the potential difference is 1.0 V?

Figure shows the 25 cm long rod of that moves to the right at a constant velocity v in a uniform perpendicular magnetic field.

a. $2 \times 10^{−19} N$ ; b. 1.25 V/m; c. 0.3125 V; d. 16 m/s

In the accompanying figure, the rails, connecting end piece, and rod all have a resistance per unit length of $2.0 Ω / cm.$ The rod moves to the left at $v equals 3.0 m / s.$ If $B equals 0.75 T$ everywhere in the region, what is the current in the circuit (a) when $a equals 8.0 cm ?$ (b) when $a equals 5.0 cm ?$ Specify also the sense of the current flow.

Figure shows the rod that slides to the left along the conducting rails at a constant velocity v in a uniform perpendicular magnetic field. Distance between the rails is 4 cm. The rod moves for the distance a.

The rod shown below moves to the right on essentially zero-resistance rails at a speed of $v equals 3.0 m/s.$ If $B equals 0.75 T$ everywhere in the region, what is the current through the $5.0 - Ω$ resistor? Does the current circulate clockwise or counterclockwise?

Figure shows the rod that slides to the right along the conducting rails at a constant velocity v in a uniform perpendicular magnetic field. Distance between the rails is 4 cm. The rails are connected through the 5 Ohm resistor.

0.018 A, CW as seen in the diagram

Shown below is a conducting rod that slides along metal rails. The apparatus is in a uniform magnetic field of strength 0.25 T, which is directly into the page. The rod is pulled to the right at a constant speed of 5.0 m/s by a force $\vec{F} .$ The only significant resistance in the circuit comes from the $2.0 - Ω$ resistor shown. (a) What is the emf induced in the circuit? (b) What is the induced current? Does it circulate clockwise or counter clockwise? (c) What is the magnitude of $\vec{F}$ ? (d) What are the power output of $\vec{F}$ and the power dissipated in the resistor?

Figure shows the rod that is pulled to the right along the conducting rails by the force F in a uniform perpendicular magnetic field. Distance between the rails is 4 cm. The rails are connected through the 2 Ohm resistor.