First slide

Faraday's law of Electromagnetic induction

Question

A square loop of side a is placed such that its plane is the same as that of a very long straight wire carrying a current I. The centre O of the loop is at a distance x from the wire. The loop is given a velocity v as shown. If x >> a, the magnitude of the emf induced in the loop is proportional to

Moderate

A

x
B

$x^{2}$
C

$\frac{1}{x}$
D

$\frac{1}{x^{2}}$

Solution

$\begin{array}{l} B = \frac{μ_{0} I}{2 π x} \\ ϕ = \frac{μ_{0} I}{2 π x} a^{2} \\ e = \frac{- d ϕ}{d t} = - \frac{μ_{0} I a^{2}}{2 π} (\frac{- 1}{x^{2}}) \frac{d x}{d t} \\ = \frac{μ_{0} I a^{2}}{2 π} v (\frac{1}{x^{2}}) \\ e α \frac{1}{x^{2}} \end{array}$

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