Question

Define mutual inductance between two long coaxial solenoids. Find out the expression for the mutual inductance of inner solenoid of length l having the radius r1 and the number of turns n1 per unit length due to the second outer solenoid of same length and r2 number of turns per unit length.

Answer 1

Mutual Inductance
The ability of production of induced emf in one coil, due to varying current in the neighbouring coil is called mutual inductance.
Magnetic flux, Φ = MI Where, M is called coefficient of mutual inductionMutual Inductance of Two Long Solenoids Consider two long solenoids S1 and S2 of same length l, such that solenoid S2 surrounds solenoid S1 completely.Φ21 = M21I1
Where, M21 is the coefficient of mutual induction of the two solenoidsMagnetic field produced inside solenoid S1 on passing current through it,B1 = μ0n1I1Magnetic flux linked with each turn of solenoid S2 will be equal to B1 times the area of cross-section of solenoid S1.Magnetic flux linked with each turn of the solenoid S2 = B1A
Therefore, total magnetic flux linked with the solenoid S2,Φ21 = B1A × n2l = μ0n1I1× A× n2lΦ21 = μ0n1n2lAI1∴ M21 = μ0n1n2AlSimilarly, the mutual inductance between the two solenoids, when current is passed through solenoid S2 and induced emf is produced in solenoid S1, is given byM12 = μ0n1n2Al∴M12 = M21 = M (say)
Hence, coefficient of mutual induction between the two long solenoids

Answer 2

Answer:

Mutual inductance, between a pair of coils, equals the magnetic flux, linked with one of them due to a change of unit current flowing in the other.

Let r

1

and r

2

be radii of inner (let it be primary P) and outer (let it be secondary S) co-axial solenoids respectively, and n

1

and n

2

be number of turns per unit length of the two solenoids. Let N

1

and N

2

be total number of turns in two solenoids and each of length l.

Let the secondary solenoid carry current I

2

. This current sets up magnetic fluxϕ

1

through the primary (inner) solenoid.

The total flux linkages with the primary solenoid are given by N

1

ϕ

1

=M

12

I

2

where , M

12

is the mutual inductance of the primary (inner) solenoid with respect to the secondary (outer) solenoid.

Magnetic field at the centre of the secondary solenoid due to a current I2 is given by B

2

=μ

0

n

2

I

2

.

The total flux linkages with the primary solenoid are given by

N

1

ϕ

1

=(n

1

l)B

2

I

1

N

1

ϕ

1

=(μ

0

πn

1

n

2

lr

1

2

)I

2

M

12

=

I

2

N

1

ϕ

1

=μ

0

πn

1

n

2

lr

1

2

Similarly

M

21

=

I

1

N

2

ϕ

2

=μ

0

πn

1

n

2

lr

1

2