Rate of change of radius through a a circular wire loop?

[zyphriss2]

A circular wire loop of radius r = 19 cm is immersed in a uniform magnetic field B = 0.690 T with its plane normal to the direction of the field. If the field magnitude then decreases at a constant rate of −1.0×10^-2 T/s, at what rate should r increase so that the induced emf within the loop is zero?



Flux=BAcos(theta)
emf=dq/dt


I tried this and I got an equation of 0=(Bcos0)(dA/dt)+(Acos0)(dB/dt)
and I plugged it into get dA/dt=.0016m^2/s and then i solved for the radius in this case which is .0228m and my answer is supposed to be in mm/s so i got 22.8 mm/s as my answer.

 

[rl.bhat]

Area = πr^2.
hence dA/dt = 2πr*dr/dt.
Now find dr/dt.

 

[nlightner]

I would like to clarify and explain the calculations and concepts used in this problem. The rate of change of radius refers to the change in the radius of the circular wire loop, which is represented by the variable r. In this problem, the wire loop has an initial radius of 19 cm or 0.19 m.

The problem states that the wire loop is immersed in a uniform magnetic field with a magnitude of 0.690 T. This means that the flux, represented by the variable Φ, passing through the loop is equal to the product of the magnetic field and the area of the loop, Φ = BA. The angle between the magnetic field and the normal to the plane of the loop is 90 degrees, so the cosine of this angle is 0.

Next, the problem states that the magnetic field magnitude is decreasing at a constant rate of −1.0×10^-2 T/s. This means that the change in the magnetic field, represented by the variable dB, is equal to -1.0×10^-2 T/s. Using the equation for flux, we can substitute in the values for B and dB to get dB/dt = -1.0×10^-2 T/s.

To find the rate at which the radius of the loop should increase in order for the induced emf to be zero, we can use the equation for emf, emf = -dΦ/dt. Since we want the emf to be zero, we can set this equation equal to zero and solve for dA/dt, which represents the rate of change of area.

Substituting in the values for Φ and dB/dt, we get -dΦ/dt = -dB/dt * A = -(-1.0×10^-2 T/s) * A = 1.0×10^-2 A/s. Since we know that Φ = BA, we can rearrange this equation to get dA/dt = 1.0×10^-2 A/s / B. Substituting in the value for B, we get dA/dt = 1.0×10^-2 A/s / 0.690 T = 1.45×10^-2 m^2/s.

Finally, to find the rate at which the radius should increase, we can use the equation for the area of a circle, A = πr^2.