An iron rod of volume `10^-4m^3` and relative permeability `1000` is placed inside a long solenoid wound with 5 turns/cm. If a current of `0*5A` is passed through the solenoid, find the magnetic moment of the rod.

Explanation

To solve the problem, we need to find the magnetic moment of an iron rod placed inside a solenoid. Here are the steps to arrive at the solution:

### Step 1: Identify Given Data
- Volume of the iron rod, \( V = 10^{-4} \, \text{m}^3 \)
- Relative permeability, \( \mu_r = 1000 \)
- Turns per centimeter of the solenoid, \( n = 5 \, \text{turns/cm} = 500 \, \text{turns/m} \)
- Current through the solenoid, \( I = 0.5 \, \text{A} \)

### Step 2: Calculate the Magnetizing Field (H)
The magnetizing field \( H \) in a solenoid is given by:
\[ H = n \cdot I \]
Substituting the values:
\[ H = 500 \, \text{turns/m} \times 0.5 \, \text{A} = 250 \, \text{A/m} \]

### Step 3: Calculate the Magnetic Field (B)
The magnetic field \( B \) in the solenoid can be expressed as:
\[ B = \mu_0 \cdot H \cdot \mu_r \]
Where \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \). Thus:
\[ B = (4\pi \times 10^{-7}) \cdot (250) \cdot (1000) \]
Calculating this:
\[ B = (4\pi \times 10^{-7}) \cdot 250000 = 3.14 \times 10^{-1} \, \text{T} \approx 0.314 \, \text{T} \]

### Step 4: Calculate the Intensity of Magnetization (I)
The intensity of magnetization \( I \) can be calculated using:
\[ I = \frac{B}{\mu_0} - H \]
Substituting the values:
\[ I = \frac{0.314}{4\pi \times 10^{-7}} - 250 \]
Calculating \( \frac{0.314}{4\pi \times 10^{-7}} \):
\[ \frac{0.314}{4\pi \times 10^{-7}} \approx 2.5 \times 10^{5} \, \text{A/m} \]
Thus:
\[ I = 2.5 \times 10^{5} - 250 \approx 2.5 \times 10^{5} \, \text{A/m} \]

### Step 5: Calculate the Magnetic Moment (M)
The magnetic moment \( M \) is given by:
\[ M = I \cdot V \]
Substituting the values:
\[ M = (2.5 \times 10^{5}) \cdot (10^{-4}) = 25 \, \text{A m}^2 \]

### Final Answer
The magnetic moment of the iron rod is \( M = 25 \, \text{A m}^2 \).

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