Two point charges are placed on the x axis. (Figure 1) The first charge, q1 = 8.00nC , is placed a distance 16.0m from the origin along the positive x axis; the second charge, q2 = 6.00nC , is placed a distance 9.00m from the origin along the negative x axis.
1. Calculate the electric field at point A, located at coordinates (0 m, 12.0m ).
The first thing to do is recognize that this is a vector type problem that requires you to set up a triangle in order to solve for the x and y components from each charge. Start by finding the lengths of each side of the triangle.
In the picture below, the side in red (from q2 to point A) has a length of 15m (use the Pythagorean theorem, e.g. sqrt(x2 + y2) = sqrt((-9)2 + 12) and the side in blue has a length of 20m. The base of the triangle is just the distance along the x-axis, which is 25m.
Next, find the angles formed by each side. The angle formed by q1 and the x-axis is 36.87ï¿½ (sin(?) = 12m / 20m, so sin-1(12/16) = 36.87ï¿½). The angle formed by q2 is 53.13ï¿½
Now find the electric field from each charge at Point A, using the formula F = kq / r2. Note that ï¿½kï¿½ is the Coulomb Constant, 8.9875 * 109 Nm2/C2. Also remember that the charges were given in nC (nano Coulombs, which is to the -9th power):
The electric field from q1 is given by:
F1 = kq1 / r12
F1 = 8.9875 * 109 * (8.00 * 10-9) / 202
F1 = 0.18 N/C
Likewise, for q2:
F2 = kq2 / r22
F2 = 8.9875 * 109 * (6.00 * 10-9) / 152
F2 = 0.24 N/C
To find the x and y components, just use trigonometry (note that the direction of the field is away from the charges, since the charges are positive, so q1ï¿½s x component will be negative, q2ï¿½s x component will be positive, and both y components will be positive:
F1, x = F1 * cos(?1)
F1, x = 0.18 * -cos(36.87)
F1, x = -0.144 N/C
F1, y = F1 * sin(?1)
F1, y = 0.18 * sin(36.87)
F1, y = 0.108 N/C
F2, x = F2 * cos(?2)
F2, x = 0.24 * cos(53.13)
F2, x = 0.144 N/C
F2, y = F2 * sin(?2)
F2, y = 0.24 * sin(53.13)
F2, y = 0.192 N/C
So the sum of the x components is 0 (+0.144 from q2, -0.144 from q1. The sum of the y components is 0.300 (0.144 + 0.192):