Faraday's Law of Induction — Question 5

Question 5

Simultaneous area growth and field decay can keep flux exactly constant

A deforming planar loop remains perpendicular to a uniform field $B(t)=B_0e^{-t/T}$. Its area A(t) can be controlled and initially equals A0.

1. Find the area history required for zero emf at every time.

2. If instead A grows linearly as $A_0(1+t/T)$, find the emf and its largest positive value for t nonnegative.

Question 5 - Solution

Simultaneous area growth and field decay can keep flux exactly constant

(a) Method and calculation

$$\mathcal E=-\frac d{dt}(BA)=0\Rightarrow BA=B_0A_0.$$$$\boxed{A(t)=A_0e^{t/T}}.$$

(b) Interpretation and further calculation

$$\Phi=B_0A_0(1+t/T)e^{-t/T}\Rightarrow\mathcal E=\frac{B_0A_0t}{T^2}e^{-t/T}.$$$$\dot{\mathcal E}\propto1-t/T\Rightarrow\boxed{\mathcal E_{\max}=B_0A_0/(eT)\text{ at }t=T}.$$

Physical check

A changing field need not cause emf if the changing geometry compensates its flux change exactly.