Problem — Ranking the EMF of four square loops

Problem

Four square loops of side $\ell$ sit in uniform fields $B$ perpendicular to their plane, which vary in time at a rate $\dd B/\dd t$:

• (A) $\ell=10\ \text{cm}$, $\dd B/\dd t=0{,}5\ \text{T/s}$
• (B) $\ell=5\ \text{cm}$, $\dd B/\dd t=1\ \text{T/s}$
• (C) $\ell=20\ \text{cm}$, $\dd B/\dd t=0{,}5\ \text{T/s}$
• (D) $\ell=10\ \text{cm}$, $\dd B/\dd t=0{,}1\ \text{T/s}$

Rank the magnitudes of the induced electromotive forces from smallest to largest.

Solution

The field is perpendicular to the plane and uniform, so the flux is $\Phi_B=B\,\ell^2$. With constant area, the induced emf is: $\abs{\text{emf}}=\ell^2\,\frac{\dd B}{\dd t}$

Calculating for each loop (sides converted to metres): $\text{(A)}\quad (0{,}10)^2\cdot 0{,}5=0{,}01\cdot 0{,}5=5\cdot 10^{-3}\ \text{V}$ $\text{(B)}\quad (0{,}05)^2\cdot 1=0{,}0025\cdot 1=2{,}5\cdot 10^{-3}\ \text{V}$ $\text{(C)}\quad (0{,}20)^2\cdot 0{,}5=0{,}04\cdot 0{,}5=2{,}0\cdot 10^{-2}\ \text{V}$ $\text{(D)}\quad (0{,}10)^2\cdot 0{,}1=0{,}01\cdot 0{,}1=1{,}0\cdot 10^{-3}\ \text{V}$

Ranking from smallest to largest: $D\ (1{,}0\cdot 10^{-3})<B\ (2{,}5\cdot 10^{-3})<A\ (5\cdot 10^{-3})<C\ (2{,}0\cdot 10^{-2})$

$\ev{D<B<A<C}$

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Topics: Electromagnetic induction Concepts: Faraday-Neumann-Lenz law · Magnetic flux Skills: Ranking reasoning