At what separation is the electrostatic force between a +13.1 µC point charge and a +26.1 µC point charge equal in magnitude to 1.77 N?

Steps to Solve

1. Understand Coulomb's Law Coulomb's law gives the formula for the electrostatic force between two point charges:

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

where:

• $F$ is the force between the charges (N)
• $k$ is Coulomb's constant, approximately $8.988 \times 10^9 , \text{Nm}^2/\text{C}^2$
• $q_1$ and $q_2$ are the magnitudes of the charges (C)
• $r$ is the separation distance (m)

2. Plug in Known Values The charges are given as:

• $q_1 = 13.1 \mu C = 13.1 \times 10^{-6} , C$
• $q_2 = 26.1 \mu C = 26.1 \times 10^{-6} , C$
• The force is given as $F = 1.77 , N$

Substituting these values into Coulomb's law:

$$ 1.77 = 8.988 \times 10^9 \frac{|(13.1 \times 10^{-6})(26.1 \times 10^{-6})|}{r^2} $$

3. Rearrange to Solve for r To find $r$, rearrange the equation:

$$ r^2 = 8.988 \times 10^9 \frac{|(13.1 \times 10^{-6})(26.1 \times 10^{-6})|}{1.77} $$

Now, calculate the value of the right-hand side.

4. Calculate the Numerical Values Calculate the numerator:

$$ (13.1 \times 10^{-6})(26.1 \times 10^{-6}) = 3.42691 \times 10^{-11} , C^2 $$

Now calculate:

$$ r^2 = 8.988 \times 10^9 \frac{3.42691 \times 10^{-11}}{1.77} $$

5. Final Calculation of r Now, calculate $r^2$ and then take the square root to get $r$:

$$ r = \sqrt{8.988 \times 10^9 \frac{3.42691 \times 10^{-11}}{1.77}} $$

Compute to find the final value of $r$.