The space inside a straight current carrying solenoid is filled with a magnetic material having magnetic susceptibility equal to `1.2xx10^(-5)`. What is factional increase in the magnetic field inside solenoid with respect to air as medium inside the solenoid?

Options

A. `1.2xx10^(-5)`
B. `1.2xx10^(-3)`
C. `1.8xx10^(-3)`
D. `2.4xx10^(-5)`

Explanation

To solve the problem of finding the fractional increase in the magnetic field inside a solenoid filled with a magnetic material, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Magnetic Field in a Solenoid:**
The magnetic field \( B_1 \) inside a solenoid filled with air (or vacuum) is given by the formula:
\[ B_1 = \mu_0 n I \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns per unit length of the solenoid,
- \( I \) is the current flowing through the solenoid.

2. **Magnetic Susceptibility:**
The magnetic susceptibility \( \chi \) of the material filling the solenoid is given as:
\[ \chi = 1.2 \times 10^{-5} \]

3. **Calculating Relative Permeability:**
The relative permeability \( \mu_r \) of the material can be calculated using the relationship:
\[ \mu_r = \chi + 1 \]
Plugging in the value of \( \chi \):
\[ \mu_r = 1.2 \times 10^{-5} + 1 = 1 + 1.2 \times 10^{-5} \]

4. **Magnetic Field with the Magnetic Material:**
The new magnetic field \( B_2 \) inside the solenoid when filled with the magnetic material is given by:
\[ B_2 = \mu_0 \mu_r n I \]

5. **Finding the Fractional Increase in Magnetic Field:**
The fractional increase in the magnetic field can be defined as:
\[ \text{Fractional Increase} = \frac{B_2 - B_1}{B_1} \]
Substituting the expressions for \( B_1 \) and \( B_2 \):
\[ \text{Fractional Increase} = \frac{\mu_0 \mu_r n I - \mu_0 n I}{\mu_0 n I} \]
Simplifying this expression:
\[ \text{Fractional Increase} = \frac{\mu_0 n I (\mu_r - 1)}{\mu_0 n I} = \mu_r - 1 \]

6. **Substituting the Value of \( \mu_r - 1 \):**
From the earlier calculation, we know:
\[ \mu_r - 1 = 1.2 \times 10^{-5} \]
Therefore, the fractional increase in the magnetic field is:
\[ \text{Fractional Increase} = 1.2 \times 10^{-5} \]

### Final Answer:
The fractional increase in the magnetic field inside the solenoid with respect to air as the medium is:
\[ \text{Fractional Increase} = 1.2 \times 10^{-5} \]