A bar magnet when placed at an angle of `30^(@)` to the direction of magnetic field induction of `5xx10^(-2)T`, experiences a moment of couple `25xx10^(-6) N-m`. If the length of the magnet is 5cm its pole strength is

Explanation

To solve the problem step by step, we will use the information provided and apply the relevant formulas.

### Step 1: Understanding the Given Data
We are given:
- Angle (θ) = 30°
- Magnetic field induction (B) = 5 x 10^(-2) T
- Moment of couple (Torque, τ) = 25 x 10^(-6) N·m
- Length of the magnet (L) = 5 cm = 5 x 10^(-2) m

### Step 2: Using the Torque Formula
The torque (τ) acting on a magnetic dipole in a magnetic field is given by the formula:
\[ \tau = mB \sin \theta \]
Where:
- m = magnetic moment
- B = magnetic field strength
- θ = angle between the magnetic moment and the magnetic field

### Step 3: Substitute the Known Values
We can rearrange the formula to solve for the magnetic moment (m):
\[ m = \frac{\tau}{B \sin \theta} \]
Substituting the known values:
- τ = 25 x 10^(-6) N·m
- B = 5 x 10^(-2) T
- sin(30°) = 0.5

Now substituting these values into the equation:
\[ m = \frac{25 \times 10^{-6}}{(5 \times 10^{-2}) \times 0.5} \]

### Step 4: Calculate the Magnetic Moment
Calculating the denominator:
\[ (5 \times 10^{-2}) \times 0.5 = 2.5 \times 10^{-2} \]
Now substituting this back into the equation for m:
\[ m = \frac{25 \times 10^{-6}}{2.5 \times 10^{-2}} = 1 \times 10^{-3} \text{ A·m}^2 \]

### Step 5: Relating Magnetic Moment to Pole Strength
The magnetic moment (m) is also related to the pole strength (p) and length (L) of the magnet by the formula:
\[ m = p \times L \]
We can rearrange this to find the pole strength (p):
\[ p = \frac{m}{L} \]

### Step 6: Substitute the Values to Find Pole Strength
Substituting the values we have:
- m = 1 x 10^(-3) A·m²
- L = 5 x 10^(-2) m

Now substituting these into the equation for p:
\[ p = \frac{1 \times 10^{-3}}{5 \times 10^{-2}} = 0.2 \times 10^{-1} = 2 \times 10^{-2} \text{ A·m} \]

### Final Answer
The pole strength of the bar magnet is:
\[ p = 2 \times 10^{-2} \text{ A·m} \]

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