Question

Four point charges `+1 muC, +1 muC, -1 muC and -1 muC` are placed at the corners A,B,C and D of a square of each side 0.1 m (i) Calculate electric potential at the center O of the square (ii) If E is middle point of BC, what is work done in carrying an electron from O to E ?

Answer 1

In fig .

`AB = BC = CD = DA = a = 0.1 m`
`BD = AC = sqrt(a^(2) + a^(2)) = a sqrt(2)`
`OA = OC = OB = OD = (1)/(2)a sqrt(2) = (a)/(sqrt(2))`
Potential at 0, `V_(0) = (1)/(4pi in_(0)) xx`
`[(1xx10^(-6))/(OA) + (1xx10^(-6))/(OB) - (1xx10^(-6))/(OC) - (1xx10^(-6))/(OD)]`
`= (10^(-6))/(4pi in_(0)) [(1)/(a// sqrt(2)) + (1)/(a// sqrt(2)) - (1)/(a// sqrt(2)) - (1)/(a// sqrt(2))]`
`= 0`
Now, `BE = CE = a//2`
Again, `AE = DE = sqrt(DC^(2) + CE^(2))`
`= sqrt(a^(2) + (a//2)^(2)) = sqrt((5a^(2))/(4) = (a sqrt(5))/(2)`
Potential at E, `V_(E) = (1)/(4pi in_(0)) xx`
`[(1xx10^(-6))/(AE) + (1xx10^(-6))/(BE) + (1xx10^(-6))/(CE) + (1xx10^(-6))/(DE)]`
`= (10^(-6))/(4pi in_(0)) [(2)/(a sqrt(5)) + (2)/(a) - (2)/(a) - (2)/(a sqrt(5))] = 0`
Work done in carrying an electron of charge `(-e)` from O to E
`W = -e [V_(E) - V_(0)] = -e [0 - 0] = zero`.