Example from Physics, 20.3 Electromagnetic Induction

EMF Induced in Conducing Coil by Moving Magnet

Imagine a magnetic field goes through a coil in the direction indicated in Figure 20.37. The coil diameter is 2.0 cm. If the magnetic field goes from 0.020 to 0.010 T in 34 s, what is the direction and magnitude of the induced current? Assume the coil has a resistance of 0.1 $Ω.$

A coil with a magnetic field running through it.

Figure 20.37 A coil through which passes a magnetic field B.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Use the equation $ε = - N Δ Φ / Δ t$ to find the induced emf in the coil, where $Δ t = 34 s$ . Counting the number of loops in the solenoid, we find it has 16 loops, so $N = 16 .$ Use the equation $Φ = B A \cos θ$ to calculate the magnetic flux
$Φ = B A \cos θ = B π {(\frac{d}{2})}^{2},$
20.30

where d is the diameter of the solenoid and we have used $\cos 0° = 1 .$ Because the area of the solenoid does not vary, the change in the magnetic of the flux through the solenoid is
$Δ Φ = Δ B π {(\frac{d}{2})}^{2} .$
20.31

Once we find the emf, we can use Ohm’s law, $ε = I R,$ to find the current.
Finally, Lenz’s law tells us that the current should produce a magnetic field that acts to oppose the decrease in the applied magnetic field. Thus, the current should produce a magnetic field to the right.
Solution
Combining equations $ε = - N Δ Φ / Δ t$ and $Φ = B A \cos θ$ gives
$ε = - N \frac{Δ Φ}{Δ t} = - N \frac{Δ B π d^{2}}{4 Δ t} .$
20.32

Solving Ohm’s law for the current and using this result gives

$\begin{matrix} I & = & \frac{ε}{R} = - N \frac{Δ B π d^{2}}{4 R Δ t} \\ = & −16 \frac{(−0.010 T) π {(0.020 m)}^{2}}{4 (0.10 Ω) (34 s)} \\ = & 15 μ A \end{matrix} .$
20.33

Lenz’s law tells us that the current must produce a magnetic field to the right. Thus, we point our right thumb to the right and curl our right fingers around the solenoid. The current must flow in the direction in which our fingers are pointing, so it enters at the left end of the solenoid and exits at the right end.

0 of 2 steps open

How did it go?

Strategy

Solution

Strategy

Solution