A short-circuited coil is placed in a time-varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled and the wire radius halved, the electrical power dissipated would be

Question

A short-circuited coil is placed in a time-varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled and the wire radius halved, the electrical power dissipated would be [2002]

Smart Achievers · Accepted Answer

(2)

$Power, P = \frac{E^{2}}{R} = \frac{π r^{2}}{ρ ℓ} {(\frac{d ϕ}{d t})}^{2} = \frac{π r^{2}}{ρ ℓ} {[\frac{d}{d t} (N B A)]}^{2}$

$[Here, E = induced emf and R = ρ \frac{ℓ}{A}]$

or, $P = \frac{π r^{2}}{ρ ℓ} N^{2} A^{2} {(\frac{d B}{d t})}^{2}$ $\Rightarrow P \propto \frac{N^{2} r^{2}}{ℓ}$

When the number of turns is quadrupled and the wire radius is halved power,

$ρ' \propto \frac{{(4 N)}^{2} {(r / 2)}^{2}}{4 ℓ}$

$∴$ $\frac{P}{ρ'} = \frac{1}{1}$

$∴$ $Power remains the same.$

Solution

Q.
A short-circuited coil is placed in a time-varying magnetic field. Electrical power is dissipated due to the current induced in the coil. If the number of turns were to be quadrupled and the wire radius halved, the electrical power dissipated would be [2002]

1 halved

2 the same

3 doubled

4 quadrupled

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