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 solenoid will be proportional to:

\[A.\,\dfrac{1}{{{L}^{2}}}\]
\[B.\,\dfrac{1}{L}\]
\[C.\,L\]
\[D.\,{{L}^{2}}\]

Answer 1

Hint: The formula that relates flux, the inductance of the coil, the magnetic field, the current flowing should be used. The product of inductance and the current is equal to the flux and in turn, the flux is equal to the number of turns, the magnetic field and the area of the coil. So, using this relationship, we will find the proportional value of the inductance of solenoid.
Formula used:
\[\begin{align}
  & \phi =NBA \\
 & \phi =LI \\
 & \phi =N{{\mu }_{0}}nI\pi {{r}^{2}} \\
\end{align}\]

Complete answer:
The formula for the flux induced in terms of the number of turns of the coil in a solenoid, the magnetc field produced and the area of the solenoid is,
\[\phi =NBA\]
Expand the above equation as follows.
The magnetic field produced is given by the formula,
\[B={{\mu }_{0}}nI\]
Where I is the current and n is the number of turns per length
\[A=\pi {{r}^{2}}\]
Where r is the radius of the solenoid.
Substitute these equations of the magnetic field and the area of the solenoid in the equation of the flux. So, we get,
\[\phi =N{{\mu }_{0}}nI\pi {{r}^{2}}\]….. (1)
The formula for the flux induced in terms of the inductance of the solenoid and the current flowing through the solenoid is,
\[\phi =LI\]…… (2)
As the LHS part of the equations (1) and (2), are the same, that is, the flux induced, so, we can equate these equations. Thus, we get,
\[N{{\mu }_{0}}nI\pi {{r}^{2}}=LI\]
Cancel out the common terms and represent the number of turns per length in terms of the number of turns of the coil.
\[N{{\mu }_{0}}\dfrac{N}{l}I\pi {{r}^{2}}=LI\]
The above equation has all the terms constant, except the inductance and the length of the solenoid. So, we have,
\[L\propto \dfrac{1}{l}\]
As the inductance of the solenoid is inversely proportional to the length of solenoid, that is, \[L\propto \dfrac{1}{l}\].

So, the correct answer is “Option B”.

Note:
Instead of the proportional value, the numerical value can be asked by providing the values of the terms. In such a case, the units of the parameters should be taken care of. All the formulae related to the flux induced in a solenoid should be known while solving such problems.