EMF Induced in a Square Loop Crossing a Finite-Width Vertical Field Region: 04-BS-9, May 2018, Q6

Question 6: EMF Induced in a Square Loop Crossing a Finite-Width Vertical Field Region (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
Loop side $L$	1 m
Velocity $v$ (westward, constant)	30 m/s
Field magnitude $B$ (vertical)	$1\times10^{-5}$ T
Field-region width (E–W, direction of travel)	30 m
Field-region extent (N–S and vertical)	infinite

Find. The induced EMF as a function of time (or position) as the loop crosses the field region — i.e. plot $\varepsilon(t)$.

Snapshot mid-entry: the loop's leading (west) edge has crossed into the 30 m-wide field band while its trailing (east) edge is still outside — only this edge sees a changing flux at this instant.

Approach. A changing flux (hence a non-zero EMF) exists only while exactly ONE edge of the loop is crossing a field boundary. Because the loop's own width (1 m) is far smaller than the field region's width (30 m), the crossing splits cleanly into three phases: entering, fully immersed, and exiting.

Entering phase. The leading edge is inside the field while the trailing edge is still outside; flux increases at the ordinary one-edge motional rate: $$\varepsilon_{\text{enter}}=BLv=(1\times10^{-5})(1)(30)$$ $$\boxed{\varepsilon_{\text{enter}}=3.000\times10^{-4}\ \text{V},\quad \text{lasting}\ \Delta t_{\text{enter}}=\frac{L}{v}=\frac{1}{30}=0.03333\ \text{s}}$$ (until the trailing edge itself reaches the field boundary).
Fully-immersed phase. Once the trailing edge also enters, the WHOLE loop sits inside a uniform field, so the enclosed flux $\Phi=BL^2$ stays constant even though the loop keeps moving: $$\varepsilon_{\text{immersed}}=0,\quad \Delta t_{\text{immersed}}=\frac{30-1}{30}=0.9667\ \text{s}$$ (this is by far the longest phase, since the 30 m region is 30× the loop's own width).
Exiting phase. Symmetric to entry: the leading edge exits first while the trailing edge is still inside, so flux now DECREASES at the same rate — same magnitude, OPPOSITE sign: $$\varepsilon_{\text{exit}}=-BLv=-3.000\times10^{-4}\ \text{V},\quad \Delta t_{\text{exit}}=\frac{L}{v}=0.03333\ \text{s}$$ After this the loop is fully clear and $\varepsilon=0$ thereafter.

EMF vs. time: a narrow +BLv pulse during entry (duration L/v), zero while fully immersed (the long middle interval), then a narrow −BLv pulse of the SAME duration during exit.

Phase	Duration	EMF
Entering ($0<t<0.03333$ s)	0.03333 s	$+3.000\times10^{-4}$ V
Fully immersed ($0.03333\ \text{s}<t<1.000$ s)	0.9667 s	$0$
Exiting ($1.000\ \text{s}<t<1.033$ s)	0.03333 s	$-3.000\times10^{-4}$ V

Concept & Theory

Topic. Motional EMF for a rigid loop crossing the boundary of a spatially bounded, finite-width uniform field region.

Key relations. $\varepsilon=-d\Phi/dt$; while exactly one edge moves through a uniform field with the rest of the loop field-free (or vice versa), $d\Phi/dt=BLv$.

Why this works. Faraday's law responds only to the NET rate of change of enclosed flux; while the loop is either wholly outside or wholly inside a uniform field, that rate is exactly zero even though the loop keeps moving, because no new field area is being enclosed or excluded. The entire EMF "story" therefore lives in the two brief transition windows when one edge alone is crossing a boundary.

Common pitfall. Assuming motion alone induces an EMF (only a CHANGE in enclosed flux does), or drawing a single continuous pulse instead of two narrow, opposite-sign pulses separated by a long zero interval.

Source. Sadiku, Elements of Electromagnetics, Ch. 9 (Faraday's law, motional EMF).

Question 6 of 8: EMF Induced in a Square Loop Crossing a Finite-Width Vertical Field Region

Question 6: EMF Induced in a Square Loop Crossing a Finite-Width Vertical Field Region (20 marks)