Maximum and Minimum EMF of a Shrinking Loop: 04-BS-9, May 2015, Q5

Question 5: Maximum and Minimum EMF of a Shrinking Loop (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Field $B$ (uniform, vertical)	$0.1$ T
Initial circumference $C_0$	$30$ cm $=0.30$ m
Rate of shrinkage $dC/dt$	$1$ cm/s $=0.01$ m/s (constant)

Find. The maximum and minimum EMF induced as the loop shrinks from $C_0$ to zero.

Loop circumference shrinks linearly from 30 cm to 0; EMF magnitude is proportional to the instantaneous circumference, so it is largest at t=0 and zero as the loop vanishes.

Approach. Express the loop area in terms of its circumference, $A=C^2/4\pi$, then apply Faraday's law $\varepsilon=-B\,dA/dt$ with $C(t)$ decreasing linearly.

Rate of change of area in terms of $C$. $$A=\frac{C^2}{4\pi}\ \Rightarrow\ \frac{dA}{dt}=\frac{C}{2\pi}\frac{dC}{dt}$$ $$|\varepsilon(t)|=B\left|\frac{dA}{dt}\right|=\frac{B\,C(t)}{2\pi}\left|\frac{dC}{dt}\right|$$ Since $|dC/dt|$ is constant, $|\varepsilon(t)|$ is directly proportional to the instantaneous circumference $C(t)$, which decreases monotonically from $C_0$ to $0$.
Maximum EMF (at $t=0$, $C=C_0$). $$\varepsilon_{\max}=\frac{B\,C_0}{2\pi}\frac{dC}{dt}=\frac{(0.1)(0.30)}{2\pi}(0.01)$$ $$\varepsilon_{\max}=\boxed{4.775\times10^{-5}\ \text{V}}$$
Minimum EMF (as $C\to0$, loop fully wound down). $$\varepsilon_{\min}=\frac{B(0)}{2\pi}\frac{dC}{dt}=\boxed{0\ \text{V}}$$

Quantity	Result
$\varepsilon_{\max}$ (at $t=0$)	$4.775\times10^{-5}$ V
$\varepsilon_{\min}$ (as $C\to0$)	$0$ V

Concept & Theory

Topic. Faraday's law for a loop of changing area at a constant magnetic field.

Key relations. $\varepsilon=-d\Phi/dt=-B\,dA/dt$; for a circle, $A=C^2/4\pi$.

Why this works. Because $B$ is constant and uniform, the only source of changing flux is the shrinking area; writing $A$ in terms of $C$ and differentiating shows the EMF magnitude scales linearly with the instantaneous circumference, since $dC/dt$ itself is held constant (a linear wind-down) — so the extremes occur exactly at the start and end of the motion, not at some interior time.

Common pitfall. Assuming $dA/dt$ is constant just because $dC/dt$ is constant — $A\propto C^2$, so $dA/dt\propto C$ and is not constant, which is precisely why the EMF has a genuine max/min rather than a single value throughout.

Source. Sadiku, Elements of Electromagnetics, Ch. 9 (Faraday's Law, motional and time-varying EMF).

Question 5 of 8: Maximum and Minimum EMF of a Shrinking Loop

Question 5: Maximum and Minimum EMF of a Shrinking Loop (20 marks)