Answer

Verified

53.1k+ views

**Hint:**Use the formula of the self-induction of the solenoid given below, substitute the formula of the length of the wire and the area of the wire in the above formula and simplify it to obtain the relation for the self-induction of the solenoid.

**Formula used:**

The self-induction is given by

$L = \dfrac{{{\mu _0}{N^2}A}}{l}$

Where $L$ is the self-induction of the solenoid, ${\mu _0}$ is the magnetic permeability, $l$ is the length of the solenoid and $A$ is the area of each turn in the solenoid.

**Complete step by step solution:**

Let us consider the wire is of length $x$

It is known that the length of the solenoid is $2\pi rN$ . Since the wire is in the shape of the cylinder, the cross sectional area is $A = \pi {r^2}$.

Use the formula of the self-induction,

$L = \dfrac{{{\mu _0}{N^2}A}}{l}$

Substitute the formula of $N = \dfrac{x}{{2\pi r}}$ and the area as $\pi {r^2}$ in the above formula.

$L = \dfrac{{{\mu _0}{{\left( {\dfrac{x}{{2\pi r}}} \right)}^2}\left( {\pi {r^2}} \right)}}{l}$

By simplifying the above equation, we get

$L = \dfrac{{{\mu _0}\left( {\dfrac{{{x^2}}}{{4{\pi ^2}{r^2}}}} \right) \times \left( {\pi {r^2}} \right)}}{l}$

By canceling the similar terms in the above step.

$L = \dfrac{{{\mu _0}\left( {\dfrac{{{x^2}}}{{4\pi }}} \right)}}{l}$

By bringing the length of the wire in the left side and other terms in the right side of the equation.

$\dfrac{{\,Ll}}{{{\mu _0}}} = \left( {\dfrac{{{x^2}}}{{4\pi }}} \right)$

By the further simplification of the above equation,

$x = \sqrt {\dfrac{{4\pi Ll}}{{{\mu _0}}}} $

Hence the length of the wire obtained is $\sqrt {\dfrac{{4\pi Ll}}{{{\mu _0}}}} $ .

**Thus the option (C) is correct.**

**Note:**The wire is in the form of the slender cylinder, hence the cross sectional area is considered as the $\pi {r^2}$ . The number of the rotation is calculated by dividing the whole length of the wire by the circumferential area of the wire as $2\pi r$.

Recently Updated Pages

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

State two factors upon which the heat absorbed by a class 10 physics JEE_Main

What will be the halflife of a first order reaction class 12 chemistry JEE_Main

Which of the following amino acids is an essential class 12 chemistry JEE_Main

Which of the following is least basic A B C D class 12 chemistry JEE_Main

Out of the following hybrid orbitals the one which class 12 chemistry JEE_Main

Other Pages

According to classical free electron theory A There class 11 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main

The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main

when an object Is placed at a distance of 60 cm from class 12 physics JEE_Main

The nitride ion in lithium nitride is composed of A class 11 chemistry JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main