EMF of an AC carrying electromagnet.

[Nev3rforev3r]

Homework Statement

This problem arises from a much larger essay/lab/project thing. I know that there is an induced current in the wire of an AC electromagnet because the magnetic field is constantly changing along with the AC current, so it makes sense to me that will produce a constantly changing emf as well which could be effectively adding or subtracting voltage of the wire in the electromagnet, which in turn slightly changes the magnetic field of the electromagnet. The larger project concerns speakers, which use an electromagnet in conjunction with a permanent magnet to oscillate the driver/cone at the same frequency as the AC current with an amplitude determined by the changing voltage of the AC current. The amplitude thing makes the EMF important. That and the lower limit of the essay word count.

Homework Equations

I didn't go quite this far into electromagnetism in my physics class, so figuring out what the relevant equations are is part of the process.

I=I₀cos(2*pi*ft)
I=V/R
B=I*mu₀/(2pi*r)
Not sure about the one directly above. Apparently it is just for a straight wire. If so I'd need something that relates the current in the wire to the magnetic field in the wire. I also have no idea how to incorporate the ferromagnetic core into the whole situation.
emf=-N*dPhi/dt

The Attempt at a Solution

I=V₀*cos(2*pi*f*t)/R
B=V₀cos(2*pi*f*t)/(2*pi*r*R)
Phi=AV₀cos(2*pi*f*t)/(2*pi*r*R)
emf=-NAV₀cos(2*pi*f*t)/(2*pi*r*R*dt)
emf=NAV₀f*sin(2*pi*f*t)/(r*R)

Didn't work. Also, this is the first time I've used legitimate mathematical calculus in a legitimate physics situation.


I still want to do something on my own, but it would be great if someone could fill the missing base pieces of "What formula should I use that relates I of a solenoid to the created B in a solenoid?" and "How do I incorporate the magnetic field of the core?"

Oh and let me know if I'm completely off in my concepts too, haha.

 

[rock.freak667]

"What formula should I use that relates I of a solenoid to the created B in a solenoid?"

Homework Equations

B=I*μ₀/(2πr)

This would be formula for the magnetic flux density due to a solenoid.

"How do I incorporate the magnetic field of the core?"

Associated with the core is a physical quantity called its magnetic permeability. μ₀ is called the permeability of a vacuum and is equal to 4π(10^-7) H/m.

So different cores have different values for μ, the higher the value of μ, the higher the value of B and hence the emf increases.

As for your derivation you have

E=-N*dΦ/dt = -N*d/dt(BA)

A does not change with time so E=-NA*dB/dt and dB/dt = dI/dt*μ₀/(2πr).

 

[Nev3rforev3r]

Ok so you're saying that
emf=-NA*dI/dt*μ0/(2πr).
The current in my situation is a regular AC current, so it changes according to
I=I₀cos(2πft)
should apply.
dI₀cos(2πft)/dt = -I₀2πf*sin(2πft)
if my math/brain is correct, that essentially gets me back to the answer I had before.

So was my end derivation correct? I said it was wrong because I graphed the equation of a current and then the derived emf formula and it seemed completely off. Maybe I did something wrong there? I don't remember exactly what numbers I used, but they seemed reasonable.

EDIT: Ahhh shoot I just realized that I've been leaving out the permeability of free space constant when deriving this stuff. Whoops.

 

[rock.freak667]

Your derivation should be correct now. Although in your answer before you did not have I₀

 

[Nev3rforev3r]

Well instead of I₀, I used V₀/R, just out of preference and the situation.

Also, I just found multiple sites that say, for a solenoid:
B=μ₀NI/l

Is that somehow equivalent to the other formula for the magnetic field of a solenoid or was there a mistake?

 

[rock.freak667]

Well instead of I₀, I used V₀/R, just out of preference and the situation.

Also, I just found multiple sites that say, for a solenoid:
B=μ₀NI/l

Is that somehow equivalent to the other formula for the magnetic field of a solenoid or was there a mistake?

Yeah that if I remember correctly is for a solenoid. Sorry about that .

 

[Nev3rforev3r]

Haha it's fine. If you think about it, the two definitions of B are weirdly equivalent. Setting them equal to each other gives you N/l=1/2πr. 2πr is the circumference of the circular loops, N is the number of loops and l is also the height of the cylinder.

Thank you for your help, though =].