Physics Faraday's Law and Induced Electric Fields

[tnbstudent]

Homework Statement

A wire loop with area 8.00 cm2 is placed inside a 20.0 cm long solenoid with 5.00×10^4 turns that carries a current of 1.10 A. The loop is concentric with the solenoid. If the length of the solenoid is stretched so that its length increases at a rate of 8.000e-2 m/s, with the number of turns remaining constant, what is the initial induced potential difference in the wire loop?

Homework Equations

Magnetic Field for Solenoid B = μ*I*n (n is number of loops per unit length)
E= ΦB/dt
Area circle = ∏r^2

The Attempt at a Solution

I know that the area of the circle is 0.0008m^2 and that is not changing
The angle is 90°
For E= ΦB/dt there are three things that can change to cause the change in B.
1- change in area so you have (dA)/dt * B* cosθ
2 - change in flux so you have A*(dB/dt)*cosθ
3 - change in the angle A*B*-sinθ

Since the question is asking about the flux through the wire loop and the change is occurring in the length of the solenoid - I do not think Area (A) is changing. I also don't think that the angle is changing. Since, the length of the solenoid is changing I think that magnetic field is what is changing as a function of time.
The formula for magnetic field is B = μ*I*n (n is number of loops per unit length). This is where I start getting messed up (at least I think this is where my troubles start)...
dB/dt=(4*∏X10^-7)*(1.1A)*[(5.0*10^4turns)/(.20m*8.0*10^-2m/s)

Another thought I had was to find B initial - which I found to be (0.346T) and then multiple it by 1/8.0*10^-2m/s so get dB/dt.

Any advice is appreciated!

Thanks

 

[tiny-tim]

hi tnbstudent!

(try using the X² and X₂ buttons just above the Reply box )

your difficulty is in finding dB/dt, when the question only gives you dL/dt (L is the length)

so write B as a function of L, and use the chain rule dB/dt = dB/dL dL/dt

(btw, are you ok now on your other solenoid question?)

 

[tnbstudent]

Thanks
This definitely gets the dL/dt in the equation which I was missing.
When I write B in terms of L I get:
B=μ*I*t/l (where t is the number of turns)
dB/dL = μ*I*t*(-1/l^2)
Is the length in this the original (.20m) or (.20m*8.0m/s *10^-2)

Our book gives answers for most questions but changes one variable. This one uses dL/dt as 3.0m/s and the answer is 4.15 *10^-4V.
When I use the information you gave me above (thanks for that - I would not have remembered to use the change rule) and I use l as .20meters I get an answer that is off by one decimal point.

Any suggestions for something I should take another look at?

(yes, i was able to get my other solenoid question - it was very easy... simply plug in the numbers but I used the wrong angle)

 

[tiny-tim]

Any suggestions for something I should take another look at?

i don't think so

(btw, it's the chain rule, and it's called that because eg dx/dt = dx/dy dy/dz dz/dw dw/dt, in a chain! )

 

[tnbstudent]

Thanks for your help. I think the book is just off by one decimal place. I did the equation using the chain rule and I got 1.11e-4 which was wrong... so I tested e-3 and it was correct.
Either way - I'm glad I understand how to do it.

Thanks again