There is one tightly wound long solenoid carrying current 2 A. Solenoid have 100 turns per cm. `H_1` is the magnetic intensity and `B_1` is the magnetic field at the centre of solenoid. Now an iron core is inserted inside the solenoid. Intensity of magnetization in the core is `4 xx 10^6` A/m. New values of magnetic intensity and magnetic field at the centre becomes `H_2` and `B_2`
Value of `B_2` is

Options

A. 10.1 T
B. 5.05 T
C. 2.05T
D. 15T

Explanation

To solve the problem step by step, we will calculate the magnetic intensity and magnetic field at the center of the solenoid both before and after inserting the iron core.

### Step 1: Calculate the Magnetic Intensity \( H_1 \)

The magnetic intensity \( H \) inside a solenoid is given by the formula:

\[ H = N \cdot I \]

Where:
- \( N \) is the number of turns per unit length (in turns/meter).
- \( I \) is the current in amperes.

Given:
- The solenoid has 100 turns per cm, which is equal to \( 100 \times 100 = 10,000 \) turns/m.
- The current \( I = 2 \) A.

Now, substituting the values:

\[ H_1 = 10,000 \, \text{turns/m} \times 2 \, \text{A} = 20,000 \, \text{A/m} = 2 \times 10^4 \, \text{A/m} \]

### Step 2: Calculate the Magnetic Field \( B_1 \) before inserting the iron core

The magnetic field \( B \) inside a solenoid is given by:

\[ B = \mu_0 H \]

Where:
- \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

Now substituting \( H_1 \):

\[ B_1 = \mu_0 H_1 = (4\pi \times 10^{-7}) \times (2 \times 10^4) \]

Calculating \( B_1 \):

\[ B_1 = 8\pi \times 10^{-3} \, \text{T} \approx 0.02513 \, \text{T} \]

### Step 3: Calculate the new Magnetic Intensity \( H_2 \) after inserting the iron core

When an iron core is inserted into the solenoid, the magnetic intensity \( H \) remains the same:

\[ H_2 = H_1 = 2 \times 10^4 \, \text{A/m} \]

### Step 4: Calculate the new Magnetic Field \( B_2 \)

The new magnetic field \( B_2 \) can be calculated using the formula:

\[ B = \mu_0 H + \mu_0 I \]

Where \( I \) is the intensity of magnetization of the core. Given that the intensity of magnetization \( I = 4 \times 10^6 \, \text{A/m} \):

Now substituting the values:

\[ B_2 = \mu_0 H_2 + \mu_0 I \]

Calculating \( B_2 \):

\[ B_2 = \mu_0 (H_2 + I) = 4\pi \times 10^{-7} \left(2 \times 10^4 + 4 \times 10^6\right) \]

Calculating the terms inside the parentheses:

\[ B_2 = 4\pi \times 10^{-7} \left(4.02 \times 10^6\right) \]

\[ B_2 = 4\pi \times 4.02 \times 10^{-1} \approx 0.5 \, \text{T} \]

### Final Answer

Thus, the value of \( B_2 \) is approximately:

\[ B_2 \approx 0.5 \, \text{T} \]

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