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

Explanation

To solve the problem, we need to analyze the relationship between the force between magnetic poles, their pole strength, and the distance between them. The formula for the force \( F \) between two magnetic poles is given by:

\[ F = \frac{K \cdot m_1 \cdot m_2}{r^2} \]

where:
- \( F \) is the force between the magnetic poles,
- \( K \) is a constant,
- \( m_1 \) and \( m_2 \) are the pole strengths of the two magnetic poles,
- \( r \) is the distance between the two poles.

### Step 1: Initial Conditions
Let the initial pole strengths be \( m_1 \) and \( m_2 \), and the initial distance be \( r \). The initial force can be expressed as:

\[ F_1 = \frac{K \cdot m_1 \cdot m_2}{r^2} \]

### Step 2: Doubling the Pole Strengths
When the pole strengths are doubled, we have:

\[ m_1' = 2m_1 \quad \text{and} \quad m_2' = 2m_2 \]

### Step 3: Doubling the Distance
When the distance is also doubled, we have:

\[ r' = 2r \]

### Step 4: New Force Calculation
Now, we can calculate the new force \( F_2 \) with the updated values:

\[ F_2 = \frac{K \cdot m_1' \cdot m_2'}{(r')^2} \]

Substituting the new values:

\[ F_2 = \frac{K \cdot (2m_1) \cdot (2m_2)}{(2r)^2} \]

### Step 5: Simplifying the New Force
Now simplify \( F_2 \):

\[ F_2 = \frac{K \cdot 4m_1 \cdot m_2}{4r^2} \]

This simplifies to:

\[ F_2 = \frac{K \cdot m_1 \cdot m_2}{r^2} \]

### Step 6: Conclusion
Thus, we find that:

\[ F_2 = F_1 \]

This means that the force between the magnetic poles remains the same even when both the pole strengths and the distance between them are doubled.

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

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