Question

Question # 9 (Mark - 2)
Figure shows two long co-axial solenoids each of length l. the outer solenoid has an area of cross section A1 and numbers of turns per unit length n1. The corresponding numbers of turns per unit length for inner solenoid is n2 and area of cross section is A2. Write their self induction L1and L2. Hence show that M < √(L1L2)

Answer 1

Answer:

ANSWER

Self-inductance of a solenoid of length L

=μ0n

2

AL, where n is the number of turns per meter.

L

1

=μ

0

n

1

2

A

1

L

and L

2

=μ

0

n

2

2

A

2

L

If I

1

is the current in outer solenoid, the magnetic field at the axis,

B

1

=μ

0

n

1

I

1

Magnetic flux linked with the secondary coil

φ=(n

2

L)B

1

A

2

=(n

2

L)((μ

0

n

1

I

1

)A

2

=μ

0

n

1

n

2

LA

2

I

1

M=

I

1

φ

=μ

0

n

1

n

2

LA

2

Now, L

1

L

2

=(μ

0

n

1

2

A

1

L),(μ

0

n

2

2

A

2

L)

L

1

L

2

=μ

0

n

1

n

2

L

A

1

A

2

L

1

L

2

M

=

A

1

A

2

As A

2

<A

1

,A

2

<

A

1

A

2

∴M<

L

1

L

2