# Does Faraday's Law Account for Varying Induced Flux in a Loop?

• kobulingam
In summary, Faraday's law states that the emf of a closed electrical circuit is determined by the external flux and the geometry and resistance of the loop. If the external flux has a nonzero second derivative, then the induced emf is changing with time, thus the induced flux has a nonzero first derivative. This varying induced flux must be considered when applying Faraday's law.
kobulingam

induced emf = - d(flux)/dt

If this is applied to a loop where induced emf causes currents, and thus flux itself, do we have to consider that flux (of course we don't if it's constant)?

If the external flux has a nonzero second derivative, then the induced emf is changing with time, thus the induced flux has a nonzero first derivative. Will this varying induced flux need to be considered when applying Faraday's law?

Yes. The internal flux must be considered except where it is much smaller than the external flux. Lenz' law describes the internal flux as opposite in direction to the external. Hence the net flux decreases. Faraday's law relates the *net* flux to the emf. Thus the emf is determined by the external flux plus the geometry and resistance of the loop itself.

Claude

This leads to the concept of "self inductance" of a coil or loop of wire.

Say we have a circular loop of wire with some area and a uniform magnetic field pointing directly into it (no angle).

What if the magnitude of B is something like

B(t) = 100T^5 + 100t^4 + 100T^3 + 100T^2 + 100T + 100

Then finding an expression for emf in loop of wire will be very hard, correct?

Because the actual flux through the loop at time t is not just Area*B'(t) , but rather (Area*B'(t) + self_flux'(t) )

Where self_flux(t) is the flux created by the loop itself.

Correct?

I worked this problem out last month, but it's at home and I'm at work right now. I'll scan it and post it later tonight.

Claude

cabraham said:
I worked this problem out last month, but it's at home and I'm at work right now. I'll scan it and post it later tonight.

Claude

I just made that question up to explain the issue I'm having in understanding Faraday's law. It's not a problem from anywhere.

If you mean that you also "considered" this issue a month ago and worked out some proof where we can ignore the self_flux, then that would be great if you can scan that work.

kobulingam said:
Because the actual flux through the loop at time t is not just Area*B'(t) , but rather (Area*B'(t) + self_flux'(t) ) Where self_flux(t) is the flux created by the loop itself.

Correct?
Isn't there a minus sign in the total flux because of Lenz' Law?

Here it is. I reuploaded it in a jpg format. I forgot about the psd format being unreadable for most. The emf, or voltage if you prefer, and current, is given by:

V = -j*omega*phi_e*R / (R + j*omega*L);

I = j*omega*phi_e / (R + j*omega*L).

Plugging in all boundary conditions makes perfect sense. If R is quite large, >> omega*L, then V reduces to:

-j*omega*phi_e, which is Faraday's law w/o considering self inductance.

Note - R = resistance of loop; L = inductance of loop; phi_e = external flux normal to loop; omega = radian frequency of time changing flux.

Claude

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Also, phi_i = internal fluz due to loop's own current.

Claude

## 1. What is Faraday's Law?

Faraday's Law states that a changing magnetic field will induce an electromotive force (EMF) in a conductor, which will in turn cause an electric current to flow.

## 2. How is induced EMF calculated?

The induced EMF can be calculated by the formula E = -N(dΦ/dt), where E is the induced EMF, N is the number of turns in the coil, and (dΦ/dt) is the rate of change of the magnetic flux through the coil.

## 3. What is Lenz's Law?

Lenz's Law is a consequence of Faraday's Law, which states that the direction of the induced current will always be such that it opposes the change that caused it. This is known as the law of conservation of energy.

## 4. What factors affect the induced EMF in a conductor?

The induced EMF in a conductor is affected by the strength of the magnetic field, the speed at which the magnetic field changes, and the number of turns in the coil.

## 5. How is Faraday's Law applied in real life?

Faraday's Law is applied in many practical applications, such as generators, transformers, and induction motors. It is also used in wireless charging technology and electromagnetic induction cooktops.

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