Two like mangetic poles of strength 10 and 40 SI units are separated by a distance 30cm. The intensity of magnetic field is zero on the line joining them

Options

A. At a point 10cm from the stronger pole
B. At a point 20cm from the stronger pole
C. At the mid point
D. At infinity

Explanation

To solve the problem, we need to find the point along the line joining two like magnetic poles where the intensity of the magnetic field is zero. The two poles have strengths of 10 and 40 SI units and are separated by a distance of 30 cm.

### Step-by-Step Solution:

1. **Identify the Magnetic Poles and Their Positions**:
- Let pole A (strength = 10) be at position 0 cm.
- Let pole B (strength = 40) be at position 30 cm.

2. **Define the Point of Interest**:
- Let point P be at a distance \( x \) from pole A. Therefore, the distance from pole B will be \( 30 - x \).

3. **Write the Expression for Magnetic Field Intensity**:
- The intensity of the magnetic field due to a magnetic pole is given by the formula:
\[ I = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^2} \]
- For pole A (strength = 10):
\[ I_1 = \frac{\mu_0}{4\pi} \cdot \frac{10}{x^2} \]
- For pole B (strength = 40):
\[ I_2 = \frac{\mu_0}{4\pi} \cdot \frac{40}{(30 - x)^2} \]

4. **Set the Intensities Equal to Each Other**:
- For the magnetic field intensity to be zero at point P, we set \( I_1 = I_2 \):
\[ \frac{10}{x^2} = \frac{40}{(30 - x)^2} \]

5. **Cross-Multiply to Solve for x**:
- Cross-multiplying gives:
\[ 10(30 - x)^2 = 40x^2 \]

6. **Expand and Rearrange the Equation**:
- Expanding the left side:
\[ 10(900 - 60x + x^2) = 40x^2 \]
\[ 9000 - 600x + 10x^2 = 40x^2 \]
- Rearranging gives:
\[ 30x^2 + 600x - 9000 = 0 \]

7. **Simplify the Quadratic Equation**:
- Dividing the entire equation by 30:
\[ x^2 + 20x - 300 = 0 \]

8. **Use the Quadratic Formula**:
- The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 1, b = 20, c = -300 \):
\[ x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot (-300)}}{2 \cdot 1} \]
\[ x = \frac{-20 \pm \sqrt{400 + 1200}}{2} \]
\[ x = \frac{-20 \pm \sqrt{1600}}{2} \]
\[ x = \frac{-20 \pm 40}{2} \]

9. **Calculate the Possible Values of x**:
- This gives:
\[ x = \frac{20}{2} = 10 \quad \text{(valid)} \]
\[ x = \frac{-60}{2} = -30 \quad \text{(not valid)} \]

10. **Determine the Distance from the Stronger Pole**:
- The distance from pole B (stronger pole) is:
\[ 30 - x = 30 - 10 = 20 \text{ cm} \]

### Final Answer:
The intensity of the magnetic field is zero at a point 20 cm from the stronger pole (pole B).