What is happens to the force between magnetic poles when their pole strenght and the distance between them both gets doubled ?

Options

A. increases to four times
B. decreases by half
C. remains unchanged
D. increases to two times

Answer

To solve the problem, we need to analyze how the force between two magnetic poles changes when both the pole strengths and the distance between them are doubled.

### Step-by-Step Solution:

1. **Understanding the Formula for Magnetic Force**:
The force \( F \) between two magnetic poles is given by the formula:
\[ F = k \frac{m_a \cdot m_b}{d^2} \]
where:
- \( F \) is the force between the poles,
- \( k \) is a constant,
- \( m_a \) and \( m_b \) are the pole strengths of the two magnets,
- \( d \) is the distance between the two poles.

2. **Initial Conditions**:
Let the initial pole strengths be \( m_a \) and \( m_b \), and the initial distance be \( d \). Thus, the initial force \( F_1 \) can be expressed as:
\[ F_1 = k \frac{m_a \cdot m_b}{d^2} \]

3. **Doubling the Pole Strengths and Distance**:
If both pole strengths are doubled, we have:
- New pole strengths: \( 2m_a \) and \( 2m_b \)
- New distance: \( 2d \)

4. **Calculating the New Force**:
The new force \( F_2 \) can be calculated using the new values:
\[ F_2 = k \frac{(2m_a) \cdot (2m_b)}{(2d)^2} \]
Simplifying this expression:
\[ F_2 = k \frac{4m_a \cdot m_b}{4d^2} = k \frac{m_a \cdot m_b}{d^2} \]

5. **Comparing the Forces**:
Now, we can compare the new force \( F_2 \) with the initial force \( F_1 \):
\[ F_2 = F_1 \]
This shows that the new force is equal to the initial force.

6. **Conclusion**:
Therefore, when both the pole strengths and the distance between them are doubled, the force between the magnetic poles remains unchanged.

### Final Answer:
The force between the magnetic poles remains unchanged when both the pole strengths and the distance between them are doubled.