Expression for induced ε from reducing Loop

[quietriot1006]

Homework Statement

A circular loop made from a flexible, conducting wire is shrinking. Its radius as a function of time is

. The loop is perpendicular to a steady, uniform magnetic field B .
Find an expression for the induced ε in the loop at time t.
Give your answer in terms of

,

, t, B and appropriate constants.

Homework Equations

Magnetic Flux (Φm)=A*B
I(induced) = ε / R
ε =Absolute value of (dΦ/dt)

The Attempt at a Solution

At first get the magnetic flux using the Area * Magnetic field. Area is (pi)r^2 * B. I use this in faradays law with dt but i don't get the write expression. I end up getting

ε(t)=(B(pi)(r_0*e^(beta*t))^2)/t

I know I am differentiating somewhere wrong but i don't know where.

 

[Redbelly98]

What's the derivative of

f(t) = e^at

 

[quietriot1006]

Wouldnt that just be a'(t)e^at?

 

[Redbelly98]

Uh, no. I meant for "a" to be a constant, sorry about the confusion.

If you can express r² in the form:

r² = r_o²e^at,​

and if you can take the derivative of e^at correctly, then you'll be able to figure out what dA/dt is.

 

[quietriot1006]

wouldn't the derivative of f(t)=e^at be just (1/a)e^at? I am not really sure and i don't understand how you were able to square the equation?

 

[Redbelly98]

No, the derivative is

a e^at​

You'll have to figure out an expression for A, using A=∏r², and differentiate it. I mentioned the whole e^at thing because the expression for A could be put in that form.

It sounds like you're having trouble remembering calculus ... has it been a while since you took it?