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 at O will the charges remain stationary ?
d. In situation (c), how much work is done in removing the charges to infinity?

Options

A. `(qQ)/(4 pi epsi_(0)r^(2))`
B. `(qQ)/(4 pi epsi_(0)(r+a)^(2))`
C. `(Qqsqrt(9r^(2) - 3a^(2)))/(4 pi epsi_(0) x^(3))`
D. `(qQ)/(2 pi in_(0) (x^(2) + a^(2))`

Answer

C

Explanation

`F=(Qq)/(4 pi e_0 r^2)` and net force on Q is `3F cos theta implies F_(net)=3. (Q q)/(4 pi e_0 r^2) cos theta`
Let G. centroid of `Delta ABC implies AB=a//sqrt3 implies h=sqrt(r^2-a^2//3)=sqrt((3r^2-a^2)/sqrt3)`
`implies cos theta =h/r=sqrt(3r^2-a^2)/(sqrt3 r)`
`implies F_(net) (sqrt3 Q q(sqrt(3r^2-a^2))/(4 pi e_0 r^3) = (Q q sqrt(9r^2-3a^2))/(4 pi e_0 r^3)`