Question

Two charged objects experience a mutual repulsive force of \(0.100 \mathrm{~N}\) If the charge of one of the objects is reduced by half and the distance separating the objects is doubled, what is the new force?

Short answer

Answer: The new force between the two charged objects is 0.0125 N.

Step-by-step solution

Write the equation for Coulomb's Law

Coulomb's Law relates the force between two charged objects to their charges and the distance between them. The equation for Coulomb's Law is: \[ F = k * \frac{q_1 * q_2}{r^2} \] where \(F\) is the force, \(k\) is the electrostatic constant, \(q_1\) and \(q_2\) are the charges of the two objects, and \(r\) is the distance between them.

Write the equation for the initial force

We are given the initial force, \(F_{initial}\), as \(0.100 N\). We can write the equation for this initial force as follows: \[ F_{initial} = k * \frac{q_1 * q_2}{r^2} \]

Write the equation for the new force

We are given that one of the charges is halved, and the distance between the objects is doubled. Let's assume the charge of the first object is halved, and the new charge is \(\frac{q_1}{2}\). The new distance is \(2r\). The equation for the new force, \(F_{new}\), will be: \[ F_{new} = k * \frac{(\frac{q_1}{2}) * q_2}{(2r)^2} \]

Solve the new force equation

To solve \(F_{new}\), first simplify the equation: \[ F_{new} = k * \frac{(\frac{q_1}{2}) * q_2}{4r^2} = \frac{1}{8} * k * \frac{q_1 * q_2}{r^2} \] Now, since we have the equation for the initial force, \(F_{initial} = k * \frac{q_1 * q_2}{r^2}\), we can find the relationship between \(F_{new}\) and \(F_{initial}\) by dividing the two equations: \[ \frac{F_{new}}{F_{initial}} = \frac{\frac{1}{8} * k * \frac{q_1 * q_2}{r^2}}{k * \frac{q_1 * q_2}{r^2}} \] Cancelling out equal terms, we get: \[ \frac{F_{new}}{F_{initial}} = \frac{1}{8} \] Now we can calculate \(F_{new}\) as a fraction of \(F_{initial}\): \[ F_{new} = \frac{1}{8} * F_{initial} \] We're given \(F_{initial} = 0.100 N\), so we can calculate \(F_{new}\): \[ F_{new} = \frac{1}{8} * 0.100 N = 0.0125 N \] The new force between the two charged objects is \(0.0125 \mathrm{~N}\).