EMF Induced in a Moving Loop Crossing a Spatially Varying Field: 04-BS-9, December 2017, Q5

Question 5: EMF Induced in a Moving Loop Crossing a Spatially Varying Field (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
Turns $N$	10
Loop area $A$	4 m$^2$
Velocity $v$ (northward, $=dx/dt$)	50 m/s
Field profile	$B(x)=B_0e^{-x/a}$, $B_0=10^{-6}$ T, $a=50$ m

Find. The induced voltage (EMF) at the instant the loop is at $x=0$.

The loop moves through a field that decays exponentially with $x$; as it moves north the enclosed flux falls, inducing an EMF even though $B$ is not explicitly time-varying.

Approach. The flux depends on time only through the loop's position, $\Phi(t)=NAB(x(t))$; apply the chain rule $d\Phi/dt=(d\Phi/dx)(dx/dt)$ with $dx/dt=v$.

Rate of change of $B$ with position at $x=0$. $$\frac{dB}{dx}=-\frac{B_0}{a}e^{-x/a}\ \Rightarrow\ \left.\frac{dB}{dx}\right|_{x=0}=-\frac{B_0}{a}=-\frac{10^{-6}}{50}=\boxed{-2.000\times10^{-8}\ \text{T/m}}$$
Induced EMF via the chain rule. $$\varepsilon=-N A\frac{dB}{dx}\Big|_{x=0}\cdot v=-(10)(4)(-2.000\times10^{-8})(50)$$ $$\boxed{\varepsilon=4.000\times10^{-5}\ \text{V}=40.00\ \mu\text{V}}$$

Quantity	Result
$dB/dx$ at $x=0$	$-2.000\times10^{-8}$ T/m
Induced EMF at $x=0$	$4.000\times10^{-5}$ V ($40.00\ \mu$V)

Concept & Theory

Topic. Motional/flux-rule EMF for a loop translating through a spatially (not temporally) varying magnetic field.

Key relations. $\varepsilon=-d\Phi/dt$, $\Phi=NAB(x)$; for motion-induced flux change, $d\Phi/dt=(d\Phi/dx)v$ by the chain rule.

Why this works. Faraday's law only requires that the FLUX through the circuit change with time; here $B$ is static in the lab frame but the loop's position $x(t)$ makes the flux it encloses time-dependent purely through its motion — the same physical effect as a motional EMF, just driven by a spatial field gradient instead of a velocity crossing a uniform field.

Common pitfall. Treating $B$ as constant (ignoring the spatial gradient) and concluding zero EMF, or differentiating with respect to $t$ directly without the chain-rule factor of $v$.

Source. Sadiku, Elements of Electromagnetics, Ch. 9 (Faraday's law, flux rule for a moving circuit).

Question 5 of 8: EMF Induced in a Moving Loop Crossing a Spatially Varying Field

Question 5: EMF Induced in a Moving Loop Crossing a Spatially Varying Field (20 marks)