Question

The total number of turns and cross section area in a solenoid is fixed. However, its length $L$ is varied by adjusting the separation between windings. The inductance of the solenoid will be proportional to:
(A). $\dfrac{1}{{{L}^{2}}}$
(B). $\dfrac{1}{L}$
(C). $L$
(D). ${{L}^{2}}$

Answer 1

Hint: The inductance is the property of a conductor by virtue of which it opposes the change in current. The flux associated with a conductor is directly proportional to the change in current. Flux also depends on the magnetic field and the area of the cross section of a conductor. Using the different relations, we can calculate the inductance of a conductor.
Formulas used:
$\phi =L'I$
$\phi =NBA$
$\Rightarrow L'=\dfrac{NBA}{I}$

Complete answer:
A coil opposes the change in current through it by virtue of a property known as inductance. The inductance is given by
$\phi =L'I$ - (1)
Here, $\phi $ is the flux passing through the coil
$L'$ is the inductance of the coil
$I$ is the current
$\phi =NBA$ - (2)
From eq (1) and eq (2), we get,
$L'I=NBA$
$\Rightarrow L'=\dfrac{NBA}{I}$ - (3)
Here, $N$ is the total number of turns in the solenoid
$A$ is the area of cross section
$I$ is the current through the solenoid
The solenoid has a magnetic field given by-
$B={{\mu }_{0}}nI$ - (4)
And the area is $A=\pi {{R}^{2}}$ - (5)
Here,
${{\mu }_{0}}$ is the permittivity of free space
$n$ is the number of turns per unit length
$I$ is the current
We substitute eq (4) and eq (5) in eq (3), we get,
$L'=\dfrac{N{{\mu }_{0}}nI\pi {{R}^{2}}}{I}$
Let the length of the solenoid be $L$. We know that, $n=\dfrac{N}{L}$ therefore,
$L'=\dfrac{{{N}^{2}}{{\mu }_{0}}\pi {{R}^{2}}}{L}$ - (6)
From the given situation, the length is variable and is changed by changing the separation between the windings.
From eq (6), the inductance of the solenoid is directly proportional to the inverse of length.
Therefore, the inductance of the solenoid is $L'=\dfrac{{{N}^{2}}{{\mu }_{0}}\pi {{R}^{2}}}{L}$ and is inversely proportional to the length.

Hence, the correct option is (B).

Note:
The change in current through a circuit results in the change in flux associated with the circuit due to which a potential is developed. The inductor is a device which opposes the change in current. The inductance of a solenoid depends on its physical properties.