A 129 129 ‑turn circular coil of radius 2.21 cm 2.21 cm and negligible resistance is immersed in a uniform magnetic field that is perpendicular to the plane of the coil. The coil is connected to a 11.7 Ω 11.7 Ω resistor to create a closed circuit. During a time interval of 0.147 s, 0.147 s, the magnetic field strength decreases uniformly from 0.637 T 0.637 T to zero. Find the energy, in millijoules, that is dissipated in the resistor during this time interval.

Electromagnetic induction occurs when there is a change in magnetic flux linkage through a coil, and an electromotive force is induced in the coil, according to Faraday-Newmann-Lenz law:

[tex]\epsilon = - \frac{N\Delta \Phi}{\Delta t}[/tex]

where

N is the number of turns in the coil

[tex]\Delta \Phi[/tex] is the change in magnetic flux through the coil

[tex]\Delta t[/tex] is the time interval

The change in magnetic flux can be rewritten as

[tex]\Delta \Phi = A\Delta B[/tex]

where

[tex]A=\pi r^2[/tex] is the area of the coil

[tex]\Delta B[/tex] is the variation of the strength of the magnetic field

So the equation becomes

[tex]\epsilon=-\frac{N\pi r^2 \Delta B}{\Delta t}[/tex]

here we have:

N = 129 turns

[tex]r=2.21 cm = 0.0221 m[/tex]

[tex]\Delta B=0-0.637 = -0.637 T[/tex]

[tex]\Delta t = 0.147 s[/tex]

So the induced emf is

[tex]\epsilon=-\frac{(129)\pi (0.0221)^2(-0.637)}{0.147}=0.857 V[/tex]

We know that the resistance of the coil is

[tex]R=11.7 \Omega[/tex]

so the current in the circuit is given by Ohm's law:

[tex]I=\frac{\epsilon}{R}=\frac{0.857}{11.7}=0.073 A[/tex]

And the power dissipated through the resistor is:

[tex]P=I^2 R=(0.073)^2(11.7)=0.062 W[/tex]

And finally, the energy dissipated in the resistor in this time interval is:

[tex]E=Pt=(0.062)(0.147)=0.00911 J = 9.11 mJ[/tex]