Faraday's Law of Induction and Current

[Kandycat]

Homework Statement

A 22.0-cm-diameter coil consists of 20 turns of circular copper wire of 2.6 mm in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of 8.65 x 10^-3 T/s. Determine (a) the current in the loop, and (b) the rate at which thermal energy is produced.

Homework Equations

V = NBA / t
R= pl/A

l = N*2*pi*diameter of coil
A = pi*(radius of wire)²

I = V/R

The Attempt at a Solution

I understand the part (a)... sort of. Tell me if I did it something wrong.

V = NBA / t
V = (20)(8.65 x 10^-3 T/s)(pi*(.0013 m)²) = 9.19 x 10^-3 V

R = pl/A
R = (1.72 x 10^-8)(20*2*pi*.22 m)/ (pi*(.0013 m)²) = 8.96 x 10^-2 ohms

I = V/R
I = 9.19 x 10^-3 V / 8.96 x 10^-2 ohms = 1.03 x 10^-5 A

Part (b) I don't understand. I don't know what I'm finding.

 

[rl.bhat]

the rate at which thermal energy is produced.
It is power and is found by P=I^2*R.

 

[nlightner]

I would like to first clarify that the equations and calculations provided by the student are correct. However, I would like to provide a more detailed explanation for part (b) of the problem.

In Faraday's Law of Induction, the changing magnetic field induces an electromotive force (EMF) in the coil, which in turn creates a current. This current then flows through the coil, generating heat due to the resistance of the wire. This heat production can be calculated using the formula P = I^2R, where P is the power (or rate of energy production), I is the current, and R is the resistance.

In this problem, we have already calculated the current (I) to be 1.03 x 10^-5 A. To find the rate of thermal energy production, we simply need to plug in the value of resistance (R) into the formula.

R = 8.96 x 10^-2 ohms

P = (1.03 x 10^-5 A)^2 * (8.96 x 10^-2 ohms) = 9.45 x 10^-10 watts

Therefore, the rate at which thermal energy is produced in the coil is 9.45 x 10^-10 watts. This value may seem small, but it is important to note that this is the rate of energy production, not the total energy produced. To find the total energy produced, we would need to know the time (t) for which the current is flowing through the coil.