Question

The force between a charge of \(25 \mu C\) and a charge of \(-10 \mu C\) is \(8.0 \mathrm{~N}\). What is the separation between the two charges? a) \(0.28 \mathrm{~m}\) c) \(0.45 \mathrm{~m}\) b) \(0.53 \mathrm{~m}\) d) \(0.15 \mathrm{~m}\)

Short answer

Answer: The separation between the two charges is 0.53 meters.

Step-by-step solution

Write down the knowns and unknowns.

In this problem, we know the following: - Charge 1: \(Q_1 = 25 \mu C\) - Charge 2: \(Q_2 = -10 \mu C\) - Force between charges: \(F = 8.0 N\) - We need to find the separation (distance) between the charges: \(r\)

Write down the formula for Coulomb's Law.

Coulomb's law describes the force between two charges and their separation: $$ F = k \frac{|Q_1 Q_2|}{r^2} $$ where \(k\) is Coulomb's constant with a value of \(8.99 × 10^9 Nm^2/C^2\), \(Q_1\) and \(Q_2\) are the charges, \(F\) is the force, and \(r\) is the separation (distance) between the charges.

Plug in the known values and solve for the unknown.

Substitute the known values into Coulomb's law and solve for the separation \(r\): $$ 8.0 N = 8.99 × 10^9 \frac{|25 \mu C (-10 \mu C)|}{r^2} $$ First, convert the microcoulombs (\(\mu C\)) to coulombs (\(C\)) by multiplying by \(10^{-6}\) $$ 8.0 N = 8.99 × 10^9 \frac{|(25 × 10^{-6})(-10 × 10^{-6})|}{r^2} $$ Now, solve for \(r\) using algebra: $$ r^2 = \frac{8.99 × 10^9 |(-250 × 10^{-12})|}{8.0 N} $$ After calculating, we find \(r^2 = 0.28125 m^2\). Take the square root of both sides to find the separation \(r\): $$ r = \sqrt{0.28125} = 0.53 m $$ Thus, the separation between the two charges is \(0.53 m\), which corresponds to choice (b) in the given options.