Question

Identical point charges \(Q\) are placed at each of the four corners of a rectangle measuring \(2.00 \mathrm{~m}\) by \(3.00 \mathrm{~m} .\) If \(Q=32.0 \mu \mathrm{C},\) what is the magnitude of the electrostatic force on any one of the charges?

Short answer

Answer: The magnitude of the electrostatic force on any one of the charges is approximately 112.56 N.

Step-by-step solution

Identify the knowns and unknowns

We are given: 1. \(Q = 32.0\ \mu\mathrm{C} = 32.0\times10^{-6}\ \mathrm{C}\) 2. The dimensions of the rectangle are \(2.00\ \mathrm{m}\) by \(3.00\ \mathrm{m}\). 3. \(k = 8.99\times10^9 \mathrm{N\ m^2/C^2}\) (Coulomb's constant) We want to find the magnitude of the electrostatic force on any one of the charges.

Calculate the force exerted by adjacent charges

Consider one of the charges at the bottom left corner of the rectangle. The other three charges are at a distance of \(2.00\ \mathrm{m}\), \(3.00\ \mathrm{m}\), and \(\sqrt{2^2 + 3^2} = \sqrt{13}\ \mathrm{m}\). First, let's calculate the force exerted by the adjacent charges at a distance of \(2.00\ \mathrm{m}\) and \(3.00\ \mathrm{m}\). For the adjacent charges, we use Coulomb's law: \(F = k \frac{Q^2}{r^2}\). 1. Force exerted by the charge at a distance of \(2.00\ \mathrm{m}\): \(F_1 = k \frac{Q^2}{(2.00)^2}\) 2. Force exerted by the charge at a distance of \(3.00\ \mathrm{m}\): \(F_2 = k \frac{Q^2}{(3.00)^2}\)

Calculate the force exerted by the diagonal charge

Now, we need to calculate the force exerted by the diagonal charge at a distance of \(\sqrt{13}\ \mathrm{m}\). Using Coulomb's law, we have: Force exerted by the charge at a distance of \(\sqrt{13}\ \mathrm{m}\): \(F_3 = k \frac{Q^2}{13}\)

Find the net force

The net force on the charge can be found by adding the vectors of the forces exerted by the other three charges. Since the forces exerted by the adjacent charges are perpendicular to each other, the net force vector acting on the charge can be found using the Pythagorean theorem: \(F_{net} = \sqrt{(F_1 + F_3\cos(\theta))^2 + (F_2 + F_3\sin(\theta))^2}\) Here, \(\theta\) is the angle between \(F_3\) and \(F_1\), which is \(180 - \arctan\left(\frac{3.00}{2.00}\right)\). Now, we can plug in the values of \(F_1\), \(F_2\), and \(F_3\): \(F_{net} = \sqrt{\left(k \frac{Q^2}{(2.00)^2} + k \frac{Q^2}{13}\cos\left(180 - \arctan\left(\frac{3.00}{2.00}\right)\right)\right)^2 + \left(k \frac{Q^2}{(3.00)^2} + k \frac{Q^2}{13}\sin\left(180 - \arctan\left(\frac{3.00}{2.00}\right)\right)\right)^2}\) Finally, we can substitute the known values of \(Q\) and \(k\): \(F_{net}\approx 112.56\ \mathrm{N}\) The magnitude of the electrostatic force on any one of the charges is roughly \(112.56\ \mathrm{N}\).