Question

There are three electric charges. Charge one is located at coordinates (−D, 0) and has a charge of +q.

Charge two is located at (D,0) and has a charge of−q.

Charge three is located at (0,H) and has a charge of +q.

(a) Find the force on charge three due to the other two charges (magnitude and direction).

(b) Imagine charge two’s charge is changed to +q. Now, find the force on charge three due to the other two charges (magnitude and direction).

Answer 1

Using Pythagorean theorem

r = distance of each charge on X-axis from the charge on y-axis = sqrt(D² + H²)

magnitude of electric force by each charge on X-axis at the location of other charge on Y-axis is given as

F = k q²/r²

F = k q²/(sqrt(D² + H²))²

F = k q²/(D² + H²)

also

Cos = D/r = D/sqrt(D² + H²)

the component of forces along the Y-direction being equal and opposite cancel out and hence we get net force along x-axis as

F_net = 2 F Cos

F_net = 2 k (q²/(D² + H²)) (D/sqrt(D² + H²))

F_net = 2k q²D/(D² + H²)^3/2

direction : towards positive X-axis

b)

Using Pythagorean theorem

r = distance of each charge on X-axis from the charge on y-axis = sqrt(D² + H²)

magnitude of electric force by each charge on X-axis at the location of other charge on Y-axis is given as

F = k q²/r²

F = k q²/(sqrt(D² + H²))²

F = k q²/(D² + H²)

also

Sin = H/r = H/sqrt(D² + H²)

the component of forces along the x-direction being equal and opposite cancel out and hence we get net force along Y-axis as

F_net = 2 F Sin

F_net = 2 k (q²/(D² + H²)) (H/sqrt(D² + H²))

F_net = 2k q²H/(D² + H²)^3/2

direction : towards positive Y-axis