# Concentric Coils EMF from Graph

#### Minescrushessouls

1. Homework Statement
Two concentric circular coils of wire lie in a plane. The larger coil has 61 turns and a radius of a = 7.40 cm. The smaller coil has 58 turns and a radius of b = 0.95 cm. The current in the larger coil has a time dependence as shown in the figure.

Picture and graph:
http://imgur.com/fHB07bx

Approximately what is the magnitude of the EMF induced in the small coil at t = 8 s?
2. Homework Equations
ξ=(d/dt)∫B⋅dA

∫B⋅dl=μ*I

3. The Attempt at a Solution

I tried taking the derivative of the slope given with respect to time
So -15*e^(-3t)
And multiplying that by the area of the small coil and the number of turns in the small coil.
I don't really know where to go from there.

I got 3.78E-12 V and thats not right

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#### TSny

Homework Helper
Gold Member
The magnetic field produced by the large coil can be considered as uniform over the area of the small coil. This simplifies calculating the flux through the small coil.

You'll need to recall how to find B at the center of a circular coil due to a current in the coil.

#### Minescrushessouls

So the Biot-Savart Law? So dB=(μIdl×r)/(4πR^2)
Can dL be simplified to 2πR meaning the law could be written as (μI)/(2R) and then multiplied by N?
But where does the smaller coil come into play? I guess I'm confused by the concept

#### TSny

Homework Helper
Gold Member
So the Biot-Savart Law? So dB=(μIdl×r)/(4πR^2)
Can dL be simplified to 2πR meaning the law could be written as (μI)/(2R) and then multiplied by N?
Yes
But where does the smaller coil come into play? I guess I'm confused by the concept
The smaller coil comes into play when you are trying to find the induced emf in the smaller coil. What do you need to know in order to find the induced emf in the smaller coil?

#### andrevdh

Homework Helper
The large coil causes a flux change in the smaller one when the current changes in it.
How are the flux change and the induced emf related?

#### Minescrushessouls

The large coil causes a flux change in the smaller one when the current changes in it.
How are the flux change and the induced emf related?
The derivative of the change in flux with respect to time is emf, but I guess I'm most confused about how to go about the induced part of this problem

#### Minescrushessouls

Yes
The smaller coil comes into play when you are trying to find the induced emf in the smaller coil. What do you need to know in order to find the induced emf in the smaller coil?
The flux with respect to time, is that where the graph comes into play?

#### TSny

Homework Helper
Gold Member
The flux with respect to time, is that where the graph comes into play?
Yes. Note that the exponential decay part of the graph is proportional to $e^{-(t-6)/2}$, not $e^{-3t}$.

#### Minescrushessouls

Yes. Note that the exponential decay part of the graph is proportional to $e^{-(t-6)/2}$, not $e^{-3t}$.
Oh ok, I must have missed that.

So it would be d/dt(μ*5e^(-(t-6)/2))/2Ra?

Does that make sense? That would be the emf in the large coil, but how do I translate that into induced emf for the smaller one? Or is it all the same?

#### TSny

Homework Helper
Gold Member
So it would be d/dt(μ*5e^(-(t-6)/2))/2Ra?
What does this expression represent?

#### Minescrushessouls

The flux of the large coil, with current as a function
When the derivative is taken with respect to time, its the emf in the large coil

#### TSny

Homework Helper
Gold Member
The flux of the large coil, with current as a function
When the derivative is taken with respect to time, its the emf in the large coil
No, it is not the flux or the time derivative of the flux.

In the expression d/dt(μ*5e^(-(t-6)/2))/2Ra, you are taking the time derivative of μ*5e^(-(t-6)/2))/2Ra. What does μ*5e^(-(t-6)/2))/2Ra represent? Refer back to post #3.

#### Minescrushessouls

No, it is not the flux or the time derivative of the flux.

In the expression d/dt(μ*5e^(-(t-6)/2))/2Ra, you are taking the time derivative of μ*5e^(-(t-6)/2))/2Ra. What does μ*5e^(-(t-6)/2))/2Ra represent? Refer back to post #3.
Oh! The magnetic field created by the large coil. So I need to multiple in dA, which would be 2*pi*r to get flux. Then I can take the time derivative to get emf

#### TSny

Homework Helper
Gold Member
Oh! The magnetic field created by the large coil.
Is it the total magnetic field created by the large coil? Remember, the large coil has a certain number of turns.
So I need to multiple in dA, which would be 2*pi*r to get flux.
What does dA represent? The formula 2*pi*r gives you the circumference of a circle.

Then I can take the time derivative to get emf
Basically, yes. But don't forget that the small coil also has a certain number of turns.

#### Minescrushessouls

Ok...

So would the correct equation be:
Na*Nb*pi*Ra^2*μ*5e^(-(t-6)/2))/2Ra

And then take the derivative with respect to time and plug in whatever value for t?

I think I understand the concept.

#### TSny

Homework Helper
Gold Member
Ok...

So would the correct equation be:
Na*Nb*pi*Ra^2*μ*5e^(-(t-6)/2))/2Ra
Sorry to be picky, but this is not an equation. It is an expression for something. Is it an expression for B? for flux? for emf?

In the expression, you have an area pi*Ra^2. This would be the area of the large coil. Is that the appropriate area to use?

You're getting close.

#### Minescrushessouls

That's the expression for flux.

I would need to use the area of the small coil because that is what experiencing the induced emf.

So Na*Nb*pi*Rb^2*μ*5e^(-(t-6)/2))/2Ra is the flux, and the time derivative will be the induced emf in the small coil

Homework Helper
Gold Member
That looks good.

#### Minescrushessouls

Thank you so much for all your help!

#### TSny

Homework Helper
Gold Member
OK, nice work. Welcome to PF.

"Concentric Coils EMF from Graph"

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