Magnetic Induction due to Changing Circuit Size

We again use Faraday’s law of induction, $\epsilon =-N\frac{\text{Δ}\text{Φ}}{\text{Δ}t}\text{,}$ although this time the magnetic field is constant and the area enclosed by the circuit changes. The circuit contains a single loop, so $N=1\text{.}$ The rate of change of the area is $\frac{\text{Δ}A}{\text{Δ}t}=v\ell \text{.}$ Thus the rate of change of the magnetic flux is

where we have used the fact that the angle $\theta$ between the area vector and the magnetic field is 0°. Once we know the emf, we can find the current by using Ohm’s law. To find the direction of the current, we apply Lenz’s law.

Faraday’s law of induction gives

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

As the rod slides to the right, the magnetic flux passing through the circuit increases. Lenz’s law tells us that the current induced will create a magnetic field that will counter this increase. Thus, the magnetic field created by the induced current must be into the page. Curling your right-hand fingers around the loop in the clockwise direction makes your right thumb point into the page, which is the desired direction of the magnetic field. Thus, the current must flow in the clockwise direction around the circuit.