What is the distance d between two point charges, where the force they exert on each other is 0.5 F when they are 20 mm apart?

Steps to Solve

1. Understanding Coulomb's Law

Coulomb's Law states that the magnitude of the force ( F ) between two point charges is given by:

$$ F = k \frac{|q_1 q_2|}{r^2} $$

where ( k ) is the electrostatic constant, ( q_1 ) and ( q_2 ) are the charges, and ( r ) is the distance between them.

2. Setting Up the Equation for Two Distances

Let’s denote the initial distance ( r_1 = d ) (the distance at which the forces are ( F )) and the modified distance ( r_2 = 20 , \text{mm} ) (where the force is ( 0.5 F )). According to Coulomb's Law:

When distance is ( r_1 ): $$ F = k \frac{|q_1 q_2|}{d^2} $$

When distance is ( r_2 ): $$ 0.5 F = k \frac{|q_1 q_2|}{(20 , \text{mm})^2} $$

3. Relating the two equations

By dividing the two equations, we find:

$$ \frac{0.5 F}{F} = \frac{k \frac{|q_1 q_2|}{(20 , \text{mm})^2}}{k \frac{|q_1 q_2|}{d^2}} $$

This simplifies to:

$$ 0.5 = \frac{d^2}{(20 , \text{mm})^2} $$

4. Solving for ( d )

Rearranging gives you:

$$ d^2 = 0.5 \times (20 , \text{mm})^2 $$

Calculating ( (20 , \text{mm})^2 ):

$$ (20 , \text{mm})^2 = 400 , \text{mm}^2 $$

Now substitute:

$$ d^2 = 0.5 \times 400 , \text{mm}^2 = 200 , \text{mm}^2 $$

Taking the square root:

$$ d = \sqrt{200 , \text{mm}^2} = 10\sqrt{2} \approx 14.14 , \text{mm} $$