Electromagnetic induction confusion

[manvin]

I was just thinking about the "changing magnetic field through a loop induces an EMF" and thought of a conceptual question I'm having trouble with. So, imagine you have an open surface where there's a changing magnetic flux that you know (say its a plane of magnetic field coming toward you changing at B=5t) now you take an arbitrary loop that does not conduct any current, and then the same arbitrary loop that does conduct current. In the first case, the EMF should be equal to dB/dT * A, or 5*A. In the second case, where current CAN flow (say through a copper conductor of resistance R) there is now an opposing magnetic flux generated by the induced current, and so the equation for the magnetic field in the center of the loop is now altered. It seems that in this case, you can no longer say that the EMF is 5*A, because the magnetic field equation is no longer B=5t since it is altered by the current inside the conductor, yet dB/dt * A = EMF still holds... Is this right? I'm sorry for the messy confusing description but if anyone can see where I'm coming from please try and answer. Thank you

 

[DrDanny]

Hi Manvin,

yes, when you have a conductor subject to a changing magnetic flux, the EMF will cause a current in that conductor. In the case of your example, you are correct that a current would be set up in your loop and that this current will generate its own magnetic field.

The current set up in the loop will create a magnetic field to counteract the applied field. However, once you reach a steady state (i.e. some time after you first started to apply the changing flux) the field generated by the current flowing in the loop will be constant. This means that the flux through the loop will keep changing at a rate of (d phi/dt) = 5 and hence the EMF calculated will be unchanged compared to your "non-conducting loop" scenario.

Things start getting a little more complex if your are interested in what happens at the beginning of the experiment, when the loop first experiences the changing flux. The time it takes for the current (and induced flux) to build up to its steady state depends on the resistance R and inductivity L of this circuit, hence it is called an LR-Circuit.
http://en.wikipedia.org/wiki/RL_circuit
http://www.electronics-tutorials.ws/inductor/lr-circuits.html

 

[Okefenokee]

Don't forget that these fields are linear. That means that they add up as simple sums.

If source A is generating magnetic flux ( we don't care what it is) and receiver B is a loop then there will be an EMF caused by A acting on B regardless of any other flux sources. Let's call it EMF_ab.

As Danny explained, the loop has some self-induction. We know that inductors will generate EMF to oppose changes in current. The receiver loop B will generate EMF that acts on receiver B. Let's call it EMF_bb.

The total EMF is then simply EMF = EMF_ab + EMF_bb. It's the superposition property of fields. I'm pretty sure that EMF_bb will turn out to be negative because of the Lorentz rule.

EMF_bb is going to be a function of EMF_ab so we can rewrite the total EMF as EMF = EMF_ab + f(EMF_ab). That function will surely be affected by the resistance of the loop don't you think? Actually we should say that the function will be affected by the impedance of the loop because the loop might be in a generator which is sending current to a load.

 

[Okefenokee]

One more thing. Be careful not to confuse units. In physics you can only add like units to each other.

All three of these are true:

EMF_total = EMF_ab + EMF_bb = EMF_ab + f(EMF_ab) all units are in Volts
PHI_total = PHI_ab + PHI_bb = PHI_ab + g(PHI_ab) this is the flux through the loop. Units are in Webers.
d/dt{PHI_total} = d/dt{PHI_ab} + d/dt{PHI_bb} = d/dt{PHI_ab} + d/dt{g(PHI_ab)} this is the change in flux through your loop. Units are in Webers per second.

 

[manvin]

Thank you DrDanny and Okefenokee, your answers were very helpful. much appreciated!