Self and Mutual Inductance in a Toroid

[libelec]

URGENT!: Self and mutual inductance

Homework Statement

1http://img20.imageshack.us/img20/8531/asdasdasv.png

a) Calculate L₁, L₂ and M for the coils in the following toroid (using thin toroid condition), the induced EMF in the second coil and the polarity.

N₁ = 500
N₂ = 200
I₁ = (20 + 0.2 t/s)A
S = 3 cm²
r_M = 5 cm
$$\mu$$_r = 1200

b) If now there's a current I₂ = 2A through coil 2 from top to bottom, and I₁ = 20A, calculate the induced EMF in each coil and the stored energy in the toroid.

The Attempt at a Solution

I can find all I'm asked for in a). There's no problem there. But in b) I can't see how there's going to be an induced EMF is the currents are continuous. I know that they actually rise from 0 to I₂ and I₁ in a period of time, but I'm not given that time. Therefor I can't use that $${\varepsilon _1} = - {L_1}\frac{{d{I_1}}}{{dt}} - {M_{21}}\frac{{d{I_2}}}{{dt}}$$ or that $${\varepsilon _2} = - {L_2}\frac{{d{I_2}}}{{dt}} - {M_{12}}\frac{{d{I_1}}}{{dt}}$$. Also, because there are two different currents going through each coil, I can't think of it in terms of an equivalent inductance L_eq.

What do I do?

 

[Andrew Mason]

I can find all I'm asked for in a). There's no problem there. But in b) I can't see how there's going to be an induced EMF is the currents are continuous.

There isn't. If dI/dt = 0 there are no induced voltage/currents. So all you have to do is determine the energy stored in the magnetic field due to the currents in each coil. This is a function of L and the current. Mutual inductance would be irrelevant.

AM

 

[libelec]

So what I have to do is to calculate U as the sum of the energies stored in each coil?

Thank you.

 

[Andrew Mason]

So what I have to do is to calculate U as the sum of the energies stored in each coil?

Think of the energy stored in the coil as the work done against the induced emf as the current builds up from 0 to the final value:

$$W = \int_0^{I_f} Pdt = \int_0^{I_f} \varepsilon i dt$$

Write out the expression for the emf in the integral for each coil to find the energy of the magnetic field for each coil. The total energy is the sum of the two integrals.

I may have overstated it in saying that mutual inductance is irrelevant because the mutual inductance does determine the induced emf as the current builds up. But I think the two mutual inductance terms cancel each other out and you are left with just the self inductance terms. Work it out and see.

AM