Calculating Induced Current in a Circular Coil with Time-Varying Magnetic Field

[ma18]

Homework Statement

A circular coil has N equal loops with diameter C. Each loop has resistivity η and diameter d.

There is a field on the central axis of B(t) = B_0 sin(ωt).

Find the induced current (ignoring other fields)

Homework Equations

emf = -d/dt * flux(t)
i = emf/R
R = η*l/A

The Attempt at a Solution

emf total = N * -d/dt ∫B dot dA = N * BA = N * B(t) * (pi*(C/2)^2)

R = η * (2*pi*d/2)/ (pi*(d/2)^2)

i = emf /R

I know I am doing something wrong and missing something, could somebody please tell me where I am going astray?

Thanks

 

[Simon Bridge]

In expanding ##R=\eta L/A## You used the diameter d in calculation for both L and A.

##L=\pi D## in this equation, D is the diameter of what?
##A=\pi D^2\!\!/4## in this equation, D is the diameter of what?

You are given two diameters labelled C and d. Which is which?

 

[ma18]

In expanding ##R=\eta L/A## You used the diameter d in calculation for both L and A.

##L=\pi D## in this equation, D is the diameter of what?
##A=\pi D^2\!\!/4## in this equation, D is the diameter of what?

You are given two diameters labelled C and d. Which is which?

C is the diameter of the coil and d is the diameter of the individual loops. I guess the equation for the length should use d while the equation for area should use C. For some reason I thought these would be equal.

 

[Simon Bridge]

C is the diameter of the coil and d is the diameter of the individual loops. I guess the equation for the length should use d while the equation for area should use C. For some reason I thought these would be equal.

I think the description is a little vague on this point to be honest - I was hoping it was consistent with other work you've done so you would know better than me by context.

Usually "coil diameter" is the diameter of the coil, which would be the same as the diameter of each loop that makes up the coil - unless we are talking about a toroidal coil perhaps. So the coil diameter and the loop diameter would be the same thing. For some reason the problem statement gives them different labels ... why would this be, unless the two labels are meant to refer to different things?

But I asked [strike]two[/strike] three questions and you have only answered one.
The D in the equation for L is different from the D in the equation for A.
You need to relate the equations to the physical dimensions of the problem.

 

[ma18]

It says more specifically that "each loop is made of a conductor with resistivity eta and conductor diameter d"

 

[Simon Bridge]

It says more specifically that "each loop is made of a conductor with resistivity eta and conductor diameter d"

... that's better.
So do you know how to finish the problem now?

You should also check the rest of your working to see if you have made similar assumptions.

 

[ma18]

I think so, because A is the cross-sectional area then d would be used whereas as L is the length of the whole wire C would be used.

 

[Simon Bridge]

Well done :)