The difference in potential of two points lying on the axis a distance of 50 cm and 100 cm respectivelym from its centre is

Options

A. `9 xx10^(7) J//Am`
B. `8xx 10^(-7) J//Am`
C. `9 xx10^(-7) J//Am`
D. `8xx 10^(7) J//Am`

Explanation

To solve the problem of finding the difference in potential (V) between two points on the axis of a magnetic dipole, we can follow these steps:

### Step 1: Understand the Formula for Potential
The potential (V) at a distance \( r \) from a magnetic dipole is given by the formula:
\[ V = \frac{\mu_0}{4\pi} \cdot \frac{m \cos \theta}{r^2} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( m \) is the magnetic moment,
- \( \theta \) is the angle between the dipole moment and the line joining the dipole to the point (for points along the axis, \( \theta = 0 \) and \( \cos \theta = 1 \)),
- \( r \) is the distance from the center of the dipole.

### Step 2: Identify the Distances
We have two points:
- Point 1 at \( r_1 = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Point 2 at \( r_2 = 100 \, \text{cm} = 1.0 \, \text{m} \)

### Step 3: Calculate the Potentials at Both Points
Using the formula for potential, we can calculate \( V_1 \) and \( V_2 \):
\[ V_1 = \frac{\mu_0}{4\pi} \cdot \frac{m \cdot 1}{(0.5)^2} \]
\[ V_2 = \frac{\mu_0}{4\pi} \cdot \frac{m \cdot 1}{(1.0)^2} \]

### Step 4: Find the Difference in Potential
The difference in potential \( \Delta V \) is given by:
\[ \Delta V = V_1 - V_2 = \left(\frac{\mu_0}{4\pi} \cdot \frac{m}{(0.5)^2}\right) - \left(\frac{\mu_0}{4\pi} \cdot \frac{m}{(1.0)^2}\right) \]
Factoring out the common terms:
\[ \Delta V = \frac{\mu_0}{4\pi} \cdot m \left(\frac{1}{(0.5)^2} - \frac{1}{(1.0)^2}\right) \]

### Step 5: Substitute the Values
Given:
- \( \mu_0 = 10^{-7} \, \text{T m/A} \)
- \( m = 3 \, \text{A m}^2 \)

Now substituting the values:
\[ \Delta V = \frac{10^{-7}}{4\pi} \cdot 3 \left(\frac{1}{0.25} - 1\right) \]
Calculating the terms:
\[ \Delta V = \frac{10^{-7}}{4\pi} \cdot 3 \left(4 - 1\right) = \frac{10^{-7}}{4\pi} \cdot 3 \cdot 3 \]
\[ \Delta V = \frac{9 \times 10^{-7}}{4\pi} \]

### Step 6: Calculate the Final Value
Using \( \pi \approx 3.14 \):
\[ \Delta V \approx \frac{9 \times 10^{-7}}{12.56} \approx 7.16 \times 10^{-8} \, \text{J/A} \]

### Final Answer
The difference in potential between the two points is approximately \( 7.16 \times 10^{-8} \, \text{J/A} \).