Understanding Faraday's Law: Solving for Induced EMF in a Uniform Magnetic Field

[lvlastermind]

Homework Statement

A circular loop of radius r = 12cm is in a uniform magnetic field B=.5T w/ its plane normal to the direction of the field. If the magnitude of B then decreases at a constant rate of -.01T/s, at rate should r increase so the induced emf in the loop is zero?

Homework Equations

Faraday's Law

The Attempt at a Solution

The question gives the initial emf in the loop = .01V and you can solve for the initial flux =.0023Wb. From here I don't know what to do.

 

[G01]

If both the radius and the field are changing, what is the formula for the induced emf?

HINT:

You should have two terms in your equation, one involving the rate of change of B and one involving the change in r. You then want to pick a value for the rate of change of r such that both terms cancel.

 

[thomate1]

I would approach this problem by first understanding the principles of Faraday's Law. This law states that a changing magnetic field will induce an electromotive force (EMF) in a conductor. In this case, the circular loop is the conductor and the changing magnetic field is caused by the decrease in B.

To solve for the induced EMF in the loop, we can use the equation E = -N*dΦ/dt, where E is the induced EMF, N is the number of turns in the loop, and dΦ/dt is the rate of change of the magnetic flux.

In this problem, we are given the initial flux and the rate of change of B. However, we need to find the rate at which the loop's radius should increase in order for the induced EMF to be zero. This can be done by setting the induced EMF equal to zero and solving for the rate of change of the loop's radius.

E = -N*dΦ/dt = 0
0 = N*dΦ/dt
0 = N*(dA/dt * dB/dt) (using the equation Φ = BA, where A is the area of the loop)
0 = N*(π*(r^2)' * (-0.01)) (substituting in the given values for A and dB/dt)
0 = N*(2πr * (-0.01)) (taking the derivative of r^2)
0 = N*(-0.02πr) (simplifying)
r = 0 (since N and π are constants, we can divide both sides by them)

Therefore, the rate of change of the loop's radius should be 0 in order for the induced EMF to be zero. This means that the loop's radius should not change at all. This makes sense, as the decrease in B is causing the induced EMF, so if we keep the radius constant, the flux will also remain constant and there will be no induced EMF.

In summary, to solve for the induced EMF in a uniform magnetic field, we can use Faraday's Law and set the induced EMF equal to zero, then solve for the rate of change of the conductor's properties (in this case, the loop's radius). This allows us to understand the relationship between the changing magnetic field and the induced EMF, and how we can manipulate the