# The magnetic field (B) inside a long solenoid having ‘n’, turns per unit length and carrying current ‘I’ when iron core is kept in it is (${{\mu }_{0}}=$ permeability of vacuum, $\chi =$ magnetic susceptibility)

A.${{\mu }_{0}}nI\left( 1-\chi \right)$

B.${{\mu }_{0}}nI\chi $

C.\[{{\mu }_{0}}n{{I}^{2}}\left( 1+\chi \right)\]

D.${{\mu }_{0}}nI\left( 1+\chi \right)$

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**Hint:**For such question, we know that magnetic field due to solenoid is given by, $\mu \times n\times I$ and relation between magnetic field and magnetic susceptibility can be given as, $\mu ={{\mu }_{0}}\left( 1+\chi \right)$, using this two formulas of magnetic field and magnetic susceptibility we will find the answer.

**Formula used:**

$B=\mu \times n\times I$, $\mu ={{\mu }_{0}}\left( 1+\chi \right)$

**Complete answer:**

In the question we are given that a long solenoid having n turns is kept in magnetic field so, first of all the equation of magnetic field of solenoid having n turns is given by,

$B=\mu \times n\times I$ …………………….(i)

Where, B is magnetic field, n is number of terms, I is current, and $\mu $ is magnetic permeability.

Now, magnetic permeability can be defined as the ratio of the magnetic induction to the magnetic intensity. In which magnetic induction is the magnetic field induced in a current carrying conductor and magnetic intensity is the intensity of magnetic field up to a certain distance.

Now, magnetic permeability can also be given by the formula,

$\mu ={{\mu }_{0}}\left( 1+\chi \right)$

Where, ${{\mu }_{0}}$is permeability of vacuum, $\chi $ is magnetic susceptibility. Magnetic susceptibility can be defined as, A measure of how much a material will become magnetized in an applied magnetic field.

Now, substituting the value of $\mu $ in expression (i) we will get,

${{B}_{\text{solenoid}}}={{\mu }_{0}}nI\left( 1+\chi \right)$

**So, the correct answer is “Option D”.**

**Additional Information:**

Magnetic susceptibility is a dimensionless proportionality constant that indicates the degree of magnetization of a material in response to an applied magnetic field.

**Note:**

The students might get confused in magnetic permeability as magnetic permeability means the property of the material to allow the magnetic line of force to pass through it and permeability of vacuum means the ability of vacuum to let the magnetic field get generated in it. As both the terms are related to each other but different as per definition they might make mistakes in substituting the value, so students must know the difference between them before solving the problem.