A rod of ferromegnetic materical with dimensions `10cmxx0.5cmxx2cm` is placed in a magnetising field of intensity `2xx10^(5)A//m`. The magnetic moment produced due it is `6` `amop-m^(2)`. The value of magnetic induction will be------`10^(-2)T`.

Options

A. 0.358T
B. 0.54T
C. 6.28T
D. 2.519T

Explanation

To solve the problem, we need to find the magnetic induction (B) produced in a ferromagnetic rod when placed in a magnetizing field. We will use the relationship between magnetic induction (B), magnetic permeability (μ₀), magnetic field intensity (H), and magnetization (I).

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Dimensions of the rod: \(10 \, \text{cm} \times 0.5 \, \text{cm} \times 2 \, \text{cm}\)
- Magnetizing field intensity: \(H = 2 \times 10^5 \, \text{A/m}\)
- Magnetic moment: \(M = 6 \, \text{A m}^2\)

2. **Calculate the Volume of the Rod:**
\[ \text{Volume} (V) = 10 \, \text{cm} \times 0.5 \, \text{cm} \times 2 \, \text{cm} = 10 \, \text{cm}^3 \]
Convert volume to cubic meters:
\[ V = 10 \times 10^{-6} \, \text{m}^3 = 10^{-5} \, \text{m}^3 \]

3. **Calculate the Intensity of Magnetization (I):**
Intensity of magnetization (I) is given by:
\[ I = \frac{M}{V} \]
Substituting the values:
\[ I = \frac{6 \, \text{A m}^2}{10^{-5} \, \text{m}^3} = 6 \times 10^5 \, \text{A/m}^2 \]

4. **Use the Formula for Magnetic Induction (B):**
The formula for magnetic induction is:
\[ B = \mu_0 H + \mu_0 I \]
where \(\mu_0\) (the permeability of free space) is:
\[ \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \]

5. **Substituting Values into the Formula:**
\[ B = \mu_0 (H + I) \]
Substitute \(H\) and \(I\):
\[ B = 4\pi \times 10^{-7} \, \text{T m/A} \times (2 \times 10^5 \, \text{A/m} + 6 \times 10^5 \, \text{A/m}) \]
\[ B = 4\pi \times 10^{-7} \times 8 \times 10^5 \]

6. **Calculate the Value:**
\[ B = 32\pi \times 10^{-2} \, \text{T} \]
Approximating \(\pi \approx 3.14\):
\[ B \approx 32 \times 3.14 \times 10^{-2} \approx 100.48 \times 10^{-2} \, \text{T} \]

7. **Final Answer:**
The value of magnetic induction \(B\) is approximately:
\[ B \approx 100.48 \times 10^{-2} \, \text{T} \]