Question

A toroidal solenoid has a mean radius rr and a cross-sectional area A and is wound uniformly with N1 turns. A second toroidal solenoid with N2 turns is wound uniformly around the first. The two coils are wound in the same direction.

Derive an expression for the inductance L1 when only the first coil is used.

Express your answer in terms of μ0, r, A, N1, and N2.

Derive an expression for the inductance L2 when only the second coil is used.

Express your answer in terms of μ0, r, A, N1, and N2.

Answer 1

(a)

we know that inductance is given as

L = N / I

where, is magnetic flux, I is current and N is number of turns

where,

= BA

where B = u_oNI / 2r

so,

for first coil, we have

L₁ = N₁ A / I ( u_oN₁I₁ / 2r)

L₁ = u_oN₁²A / 2r

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(b)

using same method and concept as (a) , we can easily write

L₂ = u_oN₂²A / 2r