Question

Three charges `q_1 = 1 mu C, q_2 = -2 mu C, and q_3 = -1 mu C` are placed at `A (0,0,0), B(-1 , 2,3,) and C (2 , -1, 1)`. Find the potential of the system of three charges at `P( 1, - 2, -1)`.

Answer 1

If `vec r_(P i)` is the position of P from the charge, its potential at P is
`V_i = (q_i)/(4 pi epsilon _0 |vec r _P - vec r _i |)`
Then, potential at P due to charge at A is
`V_1 = (q _1)/(4 pi epsilon_0 |vec r _P - vec r _A|)`
`= (10^-6 xx 10^9 xx 9)/(|(hat i -2 hat j - hat k) - (0 hat i + 0 hat j + 0 hat k)| )= (9 xx 10^3)/(sqrt (6)) V`
similarly,
`V_2 = (q _2)/(4 pi epsilon _0 | vec r _P - vec r _B|)`
`= (-2 xx 10^-6 xx 9 xx 10^9)/(|(hat i - 2 hat j - hat k) -(- hat i + 2 hat j + 3 hat k)|) = -3 xx 10^3 V`
`V_3= (q_3)/(4 pi epsilon _0 | vec r _P - vec r _C|)`
`= (-10^-6 xx 9 xx 10^9)/(|(hat i - 2 hat j - hat k)-(2 hat i - hat j + hat k)|) = -(9)/(sqrt(6)) xx 10^3 V`
Then, `V_P = Sigma V_i = V_1 + V_2 +V_3 = -3 xx 10^3 V`.