Question

A(n) 34 cm length of wire when used as a resistor has a resistance of 0.00372 Ω. The ends of the wire are connected to form a circular loop, and the plane of the loop is positioned at right angles to a uniform magnetic field that is increasing at the rate of 0.0633 T/s. At what rate is thermal energy generated in the wire? Answer in units of µW.

Answer 1

`91.1\ \mu W`

Step-by-step explanation:

P = Perimeter of loop = 34 cm

r = Radius

`(dB)/(dt)` = Rate of change of magnetic field = 0.0633 T/s

A = Area =
`\pi r^2`

R = Resistance = 0.00372 Ω

Perimeter is given by

`P=2\pi r\\\Rightarrow r=(P)/(2\pi)\\\Rightarrow r=(0.34)/(2\pi)\\\Rightarrow r=0.05411\ m`

Induced emf is given by

`\epsilon=A(dB)/(dt)\\\Rightarrow \epsilon=\pi r^2(dB)/(dt)`

Induced current is given by

`I=(\epsilon)/(R)\\\Rightarrow I=\left((1)/(R)\pi r^2(dB)/(dt)\right)`

Power is given by

`P=I^2R\\\Rightarrow P=\left((1)/(R)\pi r^2(dB)/(dt)\right)^2 R\\\Rightarrow P=(1)/(R)\left(\pi r^2(dB)/(dt)\right)^2\\\Rightarrow P=(1)/(0.00372)\left(\pi 0.05411^2* 0.0633\right)^2\\\Rightarrow P=9.11* 10^(-5)\ W=91.1\ \mu W`

The thermal energy generation rate is
`91.1\ \mu W`