Induced Voltage

Electricity and Magnetism Level 3

$\Large{x^{2}+y^{2} = t^{2}}$

A conducting loop in the $xy$ -plane takes the form of the curve given above. The parameter $t$ denotes time. Suppose that there is a uniform magnetic flux density $B$ which is normal to the $xy$ -plane.

If the magnitude of the voltage induced in the loop at time $t = 2$ can be expressed as $\alpha \pi B$ , determine the value of $\alpha$ .

Details and Assumptions:

Neglect units and assume that all physical expressions take their simplest forms.

The answer is 4.

1 solution

Steven Chase
Oct 6, 2016

The conducting loop is a circle with radius equal to $t$ .

Its area is thus $\pi t^{2}$ .

The flux linkage is equal to the magnetic flux density multiplied by the loop area, and is thus $\pi B t^{2}$ .

By Faraday's Law, the induced voltage is the time-derivative of the flux linkage, which is $2 \pi B t$ .

At $t=2$ , the induced voltage expression evaluates to $4 \pi B$ .

The constant $\alpha$ is thus 4.

Induced Voltage

The answer is 4.

1 solution

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