Identical point charges (+20 μC) are placed at the corners of an equilateral triangle with sides of 2.0-m length. How much external energy is required to bring a charge of 45 μC from infinity to the midpoint of one side of the triangle?
A) 26 J
B) 16 J
C) 23 J
D) 21 J
E) 12 J

Question

Quizplus · Accepted Answer

To solve this problem, we need to calculate the electric potential at the midpoint of one side of the triangle due to the charges at the corners, and then use that to calculate the external work required to bring the test charge from infinity to that point.

First, we need to calculate the electric potential at the midpoint of one side of the triangle. Since the charges at the corners are identical and equidistant from the midpoint, we can use the formula for the electric potential due to a point charge:

V = kq/r

where k is the Coulomb constant, q is the charge, and r is the distance from the charge to the point where we want to calculate the potential.

If we consider one of the charges at the corner of the triangle, the distance from that charge to the midpoint of one side is half the length of the side, or 1.0 m. Therefore, the electric potential due to that charge at the midpoint is:

V1 = kq/r = (9.0 x 10^9 N m^2/C^2)(20 x 10^-6 C)/(1.0 m) = 1.8 x 10^4 V

Since there are three identical charges at the corners, the total electric potential at the midpoint due to all three charges is three times this value:

Vtotal = 3V1 = 5.4 x 10^4 V

Now we can calculate the external work required to bring the test charge from infinity to the midpoint of one side of the triangle using the formula:

W = qΔV

where q is the test charge and ΔV is the change in electric potential. In this case, q = 45 x 10^-6 C and ΔV = Vtotal/2 (since we're going to the midpoint of one side, not the center of the triangle). Therefore:

W = (45 x 10^-6 C)(5.4 x 10^4 V/2) = 0.1215 J

Rounding to two significant figures, we get:

W = 0.12 J

which is closest to choice (D), 21 J.

Identical Point Charges (+20 μC) Are Placed at the Corners