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: Understand the parameters given
- Volume of the iron rod, \( V = 10^{-4} \, m^3 \)
- Relative permeability of the iron rod, \( \mu_r = 1000 \)
- Turns per unit length of the solenoid, \( n = 5 \, \text{turns/cm} = 500 \, \text{turns/m} \)
- Current through the solenoid, \( I = 0.5 \, A \)
### Step 2: Calculate the magnetic field inside the solenoid
The magnetic field \( B \) inside a solenoid is given by the formula:
\[
B = \mu_0 (H + I)
\]
Where:
- \( \mu_0 \) is the permeability of free space, \( \mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A \)
- \( H \) is the magnetic field strength, given by \( H = nI \)
First, calculate \( H \):
\[
H = n \cdot I = 500 \, \text{turns/m} \cdot 0.5 \, A = 250 \, A/m
\]
Now, substitute \( H \) into the equation for \( B \):
\[
B = \mu_0 H = (4\pi \times 10^{-7}) \cdot 250
\]
Calculating \( B \):
\[
B = 4\pi \times 10^{-7} \cdot 250 \approx 3.14 \times 10^{-4} \, T
\]
### Step 3: Calculate the magnetic permeability of the iron rod
The magnetic permeability \( \mu \) of the iron rod can be calculated using:
\[
\mu = \mu_r \cdot \mu_0
\]
Substituting the values:
\[
\mu = 1000 \cdot (4\pi \times 10^{-7}) \approx 4\pi \times 10^{-4} \, T \cdot m/A
\]
### Step 4: Calculate the intensity of magnetization \( I \)
The intensity of magnetization \( I \) can be calculated using:
\[
I = \frac{\mu - \mu_0}{\mu_0} \cdot H
\]
Substituting the values:
\[
I = \left(\frac{4\pi \times 10^{-4} - 4\pi \times 10^{-7}}{4\pi \times 10^{-7}}\right) \cdot 250
\]
Calculating \( I \):
\[
I \approx (1000 - 1) \cdot 250 \approx 999 \cdot 250 \approx 249750 \, A/m
\]
### Step 5: Calculate the magnetic moment \( M \)
The magnetic moment \( M \) is given by:
\[
M = I \cdot V
\]
Substituting the values:
\[
M = 249750 \cdot 10^{-4} \approx 24.975 \, A \cdot m^2
\]
### Final Answer
Rounding to two decimal places, the magnetic moment of the rod is approximately:
\[
M \approx 25 \, A \cdot m^2
\]
