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Self-inductance of a coaxial cable: A coaxial cable consists of a long cylinder of radius a which is surrounded by a hollow coaxial cylinder of the radius b. Find the self-induction per unit length of such a cable.

Answer
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Hint: An emf induced in a circuit due to the changes in the circuit's magnetic field is known as self-inductance. Here we have a coaxial cable consisting of two long cylinders of radius a and b. We have to use ampere’s circuital law to find the self-inductance of the coaxial cable.

Formula used:
B=μ0I2πx
where B stands for the field,
μ0 stands for the permeability of free space,
I stands for the current flowing through the circuit,
x stands for the radius of the cable.
dϕ=BdA
where dϕ stands for the induced emf,
B stands for the field, and
dA stands for the small area considered.

Complete step by step answer:
The radius of the coil inside is given as a, and the coil surrounding has a radius b.
The field in the circuit is given by,
B=μ0I2πx
The emf induced in a small area is given by,
dϕ=BdA
Substituting the value of B in the above equation, we get
dϕ=μ0I2πxldx (Considering a small rectangular area, dA=ldx)

To find the total induction, we have to integrate the above equation between the radii of the two cables, a and b.
dϕ=abμ0I2πxldx
Taking the constants out,
dϕ=μ0Il2πabdxx
Integrating, we get
ϕ=μ0Il2π[lnx]ab
Applying the limits, we get
ϕ=μ0Il2π[lnblna]
This can be written as,
ϕ=μ0Il2π[lnblna]

When a current passes through a coil, flux is associated with the coil. The flux ϕis proportional to the current I through the coil.
That is,
ϕI
This can be written as,
ϕ=LI
where L is called the coefficient of self-induction or self-inductance.

Substituting this value for ϕ,
LI=μ0Il2πlnba
Common terms on both sides are cancelled, we get
L=μ0l2πlnba
The self-inductance of a coaxial coil is thus, L=μ0l2πlnba

Note: The phenomenon by which a coil opposes the growth or decay of current through it by producing an emf. When 1A current passes through the coil, the coefficient of self-induction of a coil is numerically equal to the flux linked with the coil.
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