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

Options

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

Explanation

To solve the problem of finding the point where the intensity of the magnetic field is zero between two like magnetic poles, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Strength of the first magnetic pole (m1) = 10 SI units
- Strength of the second magnetic pole (m2) = 40 SI units
- Distance between the two poles (d) = 30 cm = 0.3 m

2. **Set Up the Problem**:
- Let the distance from the first pole (10 SI units) to the point where the magnetic field is zero be \( x \).
- Therefore, the distance from the second pole (40 SI units) to the same point will be \( (30 - x) \) cm.

3. **Write the Expression for Magnetic Fields**:
- The magnetic field \( B_1 \) due to the first pole at distance \( x \) is given by:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{m_1}{x^2} = \frac{\mu_0}{4\pi} \cdot \frac{10}{x^2} \]
- The magnetic field \( B_2 \) due to the second pole at distance \( (30 - x) \) cm is given by:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{m_2}{(30 - x)^2} = \frac{\mu_0}{4\pi} \cdot \frac{40}{(30 - x)^2} \]

4. **Set the Magnetic Fields Equal to Each Other**:
- Since the point where the magnetic field is zero is where the two magnetic fields are equal in magnitude, we set \( B_1 = B_2 \):
\[ \frac{10}{x^2} = \frac{40}{(30 - x)^2} \]

5. **Cross Multiply and Simplify**:
- Cross multiplying gives:
\[ 10(30 - x)^2 = 40x^2 \]
- Expanding the left side:
\[ 10(900 - 60x + x^2) = 40x^2 \]
\[ 9000 - 600x + 10x^2 = 40x^2 \]
- Rearranging the equation:
\[ 30x^2 + 600x - 9000 = 0 \]

6. **Solve the Quadratic Equation**:
- Dividing the entire equation by 30:
\[ x^2 + 20x - 300 = 0 \]
- Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot (-300)}}{2 \cdot 1} \]
\[ x = \frac{-20 \pm \sqrt{400 + 1200}}{2} \]
\[ x = \frac{-20 \pm \sqrt{1600}}{2} \]
\[ x = \frac{-20 \pm 40}{2} \]
- This gives us two solutions:
\[ x = 10 \quad \text{(valid, as it is between the poles)} \]
\[ x = -30 \quad \text{(not valid, as distance cannot be negative)} \]

7. **Calculate the Distance from Each Pole**:
- The distance from the first pole (10 SI units) is \( x = 10 \) cm.
- The distance from the second pole (40 SI units) is \( 30 - x = 30 - 10 = 20 \) cm.

### Final Answer:
The intensity of the magnetic field is zero at a point 10 cm from the first pole and 20 cm from the second pole.

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