Question

For a solenoid keeping the turn density constant its length makes halved and its cross section radius is doubled then the inductance of the solenoid increased by :-
A. \[200\% \]
B. \[100\% \]
C. \[800\% \]
D. \[700\% \]

Answer 1

Hint:A solenoid is a coil of wire generally in the cylindrical shape and acts as the magnet when carried the electric current through it. The inductance of the solenoid can be expressed by $L = \dfrac{{{\mu _o}{N^2}A}}{l}$ where ${\mu _o}$ is the magnetic constant, N is the number of turns, $I$ is the current and $l$ is the length of the solenoid and A is area of cross-section. Here we will use the standard formula and place the given conditions in it and simplify.

Complete step by step solution:
The inductance of the solenoid,$L = \dfrac{{{\mu _o}{N^2}A}}{l}$ ....(i)
Given that- New length of the solenoid becomes halved of the original length.
$L' = \dfrac{L}{2}$ .... (A)
Radius is doubled.
$r' = 2r$ ..... (B)
Now, the new inductance of a solenoid can be given by-
$L' = \dfrac{{{\mu _o}{N^2}\pi r{'^2}}}{{l'}}$
Place the values from equation (A) and (B)
\[L' = \dfrac{{{\mu _o}{N^2}\pi {{(2r)}^2}}}{{\dfrac{l}{2}}}\]

Denominator’s denominator goes to the numerator of the fraction –
\[L' = \dfrac{{{\mu _o}{N^2}\pi 4{r^2} \times 2}}{l}\]
Simplify the above equation –
\[L' = \dfrac{{8{\mu _o}{N^2}\pi {r^2}}}{l}{\text{}}.....{\text{ (ii)}}\]
Take ratio of the equations (i) and (ii)
\[\dfrac{{L'}}{L} = \dfrac{{\dfrac{{8{\mu _o}{N^2}\pi {r^2}}}{l}}}{{\dfrac{{{\mu _o}{N^2}\pi {r^2}}}{l}}}{\text{}}\]
Same terms cancel each other. So remove them from the numerator and the denominator.
\[\therefore\dfrac{{L'}}{L} = 8{\text{}}\]
Thus, we can say that there is eight times increase in the inductance means the inductance is increased by $800\% $

Hence, the option C is the correct answer.

Additional information:
Solenoid can be expressed as - $B = {\mu _o}\dfrac{{NI}}{l}$ where B is the solenoid magnetic flux density, ${\mu _o}$ is the magnetic constant, N is the number of turns, $I$ is the current and $l$ is the length of the solenoid.

Note: Remember the correct formula and simplify the given conditions in the mathematical form and then substitute in the formula. Be careful while taking the ratio of the inductance and change in the inductance. Also, know that if it is increased eight times means it is increased $800$ times from the original.