The point charges + q ,- q ,- q ,+ q ,+ Q and -q are placed at the vertices of a regular hexagon ABCDEF as shown in figure. The electric field at the centre of hexagon ' O ' due to the five charges at A, B, C, D and F is twice the electric field at centre ' O ' due to charge +Q at E alone. The value of Q is

Step-by-step solution

The electric field at O due to charge q' at distance r is E = kq' r^2 where k = 1 4 ₀ , directed radially from the charge. For a regular hexagon, the vectors from the center to the vertices satisfy _ i=1 ^6 u _i = 0 , so u _A + u _B + u _C + u _D + u _F = - u _E . The combined field from the five charges at A, B, C, D, and F is therefore E _ total,5 = kq r^2 (- u _E) , with magnitude kq r^2 directed opposite to u _E . The field from charge +Q at E alone is E _ E = kQ r^2 u _E with magnitude kQ r^2 . Given that | E