A magnetic dipole has a pole strength of 40 Am and magnetic length 20 cm. What is the magnetic induction at a point on its axis at a distance of 30 cm from the centre of the dipole?
`(mu_(0)/(4pi)=10^(-7)" Wb/Am")`

Options

A. `4.5 xx10^(-5)" Wb/m"^(2)`
B. `5.5 xx10^(-5)" Wb/m"^(2)`
C. `6.5 xx10^(-5)" Wb/m"^(2)`
D. `7.5 xx10^(-5)" Wb/m"^(2)`

Explanation

To find the magnetic induction at a point on the axis of a magnetic dipole, we can use the formula for the magnetic induction \( B \) at a distance \( d \) from the center of the dipole:

\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2m}{d^3} \]

Where:
- \( \mu_0 / 4\pi = 10^{-7} \, \text{Wb/Am} \)
- \( m \) is the magnetic moment of the dipole
- \( d \) is the distance from the center of the dipole to the point where we want to find the magnetic induction.

### Step 1: Calculate the magnetic moment \( m \)
The magnetic moment \( m \) is given by:

\[ m = m_p \cdot l \]

Where:
- \( m_p \) is the pole strength
- \( l \) is the length of the dipole

Given:
- Pole strength \( m_p = 40 \, \text{Am} \)
- Magnetic length \( 2l = 20 \, \text{cm} = 0.2 \, \text{m} \), thus \( l = 0.1 \, \text{m} \)

Now, we can calculate \( m \):

\[ m = 40 \, \text{Am} \cdot 0.1 \, \text{m} = 4 \, \text{Am}^2 \]

### Step 2: Convert the distance \( d \) to meters
The distance from the center of the dipole to the point where we want to find the magnetic induction is given as \( 30 \, \text{cm} \):

\[ d = 30 \, \text{cm} = 0.3 \, \text{m} \]

### Step 3: Substitute the values into the formula for \( B \)
Now we can substitute \( m \) and \( d \) into the formula for \( B \):

\[ B = \frac{10^{-7}}{4\pi} \cdot \frac{2 \cdot 4}{(0.3)^3} \]

Calculating \( (0.3)^3 \):

\[ (0.3)^3 = 0.027 \]

Now substituting this back into the equation for \( B \):

\[ B = \frac{10^{-7}}{4\pi} \cdot \frac{8}{0.027} \]

### Step 4: Calculate the numerical value
Using \( \pi \approx 3.14 \):

\[ B = \frac{10^{-7}}{12.56} \cdot \frac{8}{0.027} \]

Calculating \( \frac{8}{0.027} \):

\[ \frac{8}{0.027} \approx 296.296 \]

Now substituting this back into the equation for \( B \):

\[ B \approx \frac{10^{-7}}{12.56} \cdot 296.296 \approx 2.36 \times 10^{-6} \, \text{Wb/m}^2 \]

### Final Step: Convert to appropriate units
To express this in terms of \( \text{Am}^{-2} \):

\[ B \approx 2.36 \times 10^{-6} \, \text{T} \text{ or } 2.36 \times 10^{-5} \, \text{Wb/m}^2 \]

### Conclusion
Thus, the magnetic induction at the point on the axis at a distance of 30 cm from the center of the dipole is approximately:

\[ B \approx 7.5 \times 10^{-5} \, \text{T} \]