Three charges each of value q are placed at the corners of an equilateral triangle. A fourth charge Q is placed at the center of the triangle.
a. Find the net force on charge q.
b. If `Q = -q`, will the charges at the corners move toward the center or fly away from it ?
c. For what value of Q and O will the charges remain stationary ?
d. In situation (c), how much work is done in removing the charges to infinity ?

Answer

a. `F_R = q/(4 pi epsilon_0 a^2) [sqrt(3) q + 3 Q]` along `vec(O A)`
b. the charges `q` at corners A,B and C will move towards the center `O`.
c. `Q = -(q/sqrt(3))`
d. zero

Explanation

A If `Q` is a positive charge, the resultant force on q at A will be
`vec F _R = vec F _1 + vec F_2 + vec F_3`
Here, `F_1 = F_2 = (1)/(4 pi epsilon_0) (q^2)/(a^2)`
So, `(vec F_1 + vec F_2) = 2 xx (1)/(4 pi epsilon_0) (q^2)/(a^2) cos 30^(@) = (sqrt(3 q^2))/(4 pi epsilon_0 a^2)` along `vec(O A)`
while `vec F_3 = (1)/(4 pi epsilon_0) (q Q)/((a // sqrt(3))^2) = (3 q Q)/(4 pi epsilon_0 a^2)` along `vec( O A)`
.
So, `F_R = q/(4 pi epsilon_0 a^2) [sqrt(3) q + 3 Q]` along `vec(O A)` .... (i)
b. For `Q = -q`,
`F_R = q/(4 pi epsilon_0 a^2) [sqrt(3)q - 3 a]` along `vec(O A)`
=`(sqrt(3) q)/(4 pi epsilon _0 a^2) [sqrt(3) - 1]` along `vec(A O)`
i.e., the charges `q` at corners A, B and C will move toward the center O.
c. Charges will remain stationary if `F_R = 0`, which Eq. (i) is possible only if
`[(sqrt(3)) q + 3 Q] = 0`, i,e., `Q = -(q/sqrt(3))`
d. Potential energy of the system
`U = (1)/(2) xx (1)/(4 pi epsilon_0) sum_(i ne j)(q_(i)q_(j))/r_(ij) = (1)/(4 pi epsilon_0) [3(q xx q)/a + (3 q(- q // sqrt(3)))/((a//sqrt(3)))] = 0`
And for finite charges, distribution potential energy at infinity is always zero. So
`W = U_F - U_I= 0 - 0 = 0`.