A) Calculate the resultant electric field strength at the midpoint between the charges.
Qx is the charge at X and Qy is the charge at Y.
E at midpoint = k×Qx/0.25² - k×Qy/0.25²
k = 9×10⁹Nm²C⁻², Qx = 45nC, Qy = 12nC
E = 4752N/C
Well done.
B) Calculate the distance from X at which the electric field strength is zero.
Let D be some point between X and Y for which the net E field is 0.
Let d be the distance from X to D.
Set up the following equation:
E at D = k×Qx/d² - k×Qy/(0.5-d)² = 0
Do some algebra to solve for d:
k×Qx/d² = k×Qy/(0.5-d)²
Qx/d² = Qy/(0.5-d)²
Qx(0.5-d)² = Qyd²
(0.5-d)√Qx = d√Qy
0.5√Qx-d√Qx = d√Qy
d(√Qx+√Qy) = 0.5√Qx
d = (0.5√Qx)/(√Qx+√Qy)
Plug in Qx = 45nC, Qy = 12nC
d ≈ 330mm
C) Calculate the magnitude of the electric field strength at the point P on the diagram below.
First determine the angles of the triangle. The sides of the triangle are 0.3m, 0.4m, and 0.5m, so this is a right triangle where the angle between the 0.3m and 0.4m sides is 90°
∠Y = tan⁻¹(0.4/0.3) = 53.13°
∠X = 90-∠Y = 36.87°
Determine the horizontal component of E at P:
Ex = E from Qx × cos(∠X) - E from Qy × cos(∠Y)
Ex = k×Qx/0.4²×cos(36.87°) - k×Qy/0.3²×cos(53.13°)
Ex = 1305N/C
Determine the vertical component of E at P:
Ey = E from Qx × sin(∠X) - E from Qy × sin(∠Y)
Ey = k×Qx/0.4²×sin(36.87°) - k×Qy/0.3²×sin(53.13°)
Ey = 2479N/C
Use the Pythagorean theorem to determine the magnitude of E at P:
E = √(Ex²+Ey²)
E ≈ 2802N/C