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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-2/16d/b/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 16D / b / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

t

, let the centre of the loop have coordinate

Y = d + v t

. With normal

+ e_{x}

, the magnetic flux is

Φ_{B} = - \int_{Y - a}^{Y + a} \int_{- a}^{a} \frac{μ _{0} I}{2 π y}, d z, d y = - \frac{μ _{0} I a}{π} lo g \frac{Y + a}{Y - a} .

(1)

Faraday's law and Ohm's law therefore give a current of magnitude

∣ I_{loop} (t) ∣ = \frac{2 μ _{0} I a ^{2} v}{π R ( ( d + v t ) ^{2} - a ^{2} )} .

(2)

The original flux points into the plane and decreases in magnitude as the loop recedes. By Lenz's law, the induced current reinforces the into-plane field: it is clockwise when viewed from the

+ x

side.

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