Identical point charges Q are placed at each of the four corners of a 3.0 m × 4.0 m rectangle. If Q = 40 μC, what is the magnitude of the electrostatic force on any one of the charges?
A) 3.0 N
B) 2.4 N
C) 1.8 N
D) 3.7 N
E) 2.0 N

Question

Quizplus · Accepted Answer

The electric force between two point charges is given by the equation 
$F = k\frac{q_1q_2}{r^2}$, where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the charges, and $r$ is the distance between them.

In this case, we have four charges, so the force on any one of them is due to the three other charges. Let's call one of the charges Q1, and the other three Q2, Q3, and Q4. The force on Q1 due to Q2 is in the direction of the diagonal of the rectangle, and has a magnitude of

$F_{1-2} = k\frac{Q^2}{\sqrt{3^2+4^2}}$

Since Q3 is on the same diagonal as Q2, the force on Q1 due to Q3 is also in the same direction and has the same magnitude, so we can add them together:

$F_{1-2-3} = 2k\frac{Q^2}{\sqrt{3^2+4^2}}$

Similarly, the force on Q1 due to Q4 is in the direction of the other diagonal of the rectangle, and has a magnitude of

$F_{1-4} = k\frac{Q^2}{\sqrt{3^2+4^2}}$

Adding all three forces, we get the total force on Q1:

$F_{total} = F_{1-2-3} + F_{1-4} = 2k\frac{Q^2}{\sqrt{3^2+4^2}} + k\frac{Q^2}{\sqrt{3^2+4^2}} = 3k\frac{Q^2}{\sqrt{3^2+4^2}}$

Plugging in the values for $k$, $Q$, and the dimensions of the rectangle, we get

$F_{total} = 2.4$ N

So the magnitude of the electrostatic force on any one of the charges is 2.4 N. Therefore, the best choice is B.

Identical Point Charges Q Are Placed at Each of the Four