Can you arrange three charges such that there exist a point where the electric p

Question

Can you arrange three charges such that there exist a point where the electric potential is zero, but not the electric field? Arrange four charges such that the E-field is zero at the central point, but not the electric potential. For your configuration in (2a) calculate the electric potential. Calculate the electric potential energy of your configuration. Arrange four charges in such a way that both, the electric field and the electric potential, are zero at the same point. For your configuration in (3a) calculate the electric potential energy.

Explanation / Answer

problem 1:

three charge -q, -q and 2*q are placed on the corners of a equilateral triangle,

then at the centroid of the triangle, potential is zero but electric field is non-zero.

problem 2:

let there be a square with coordinates:

(0,0) -> charge placed =Q
(d,0) -> charge placed=-2*Q
(d,d) -> charge placed=Q
(0,d) -> charge placed=-2*Q

due to symmetry of diamterically opposite charges,

net field will be zero.

but due to mismatch of magnitude of charges at adjacent corners,

net potential will be non-zero

part b:

center point is at a distance of sqrt(2)*d/2 from each corner

then net electric potential=(k/(d/sqrt(2)))*(Q+Q-2*Q-2*Q)

=-2*sqrt(2)*k*Q/d

part c:

electric potential energy of the configuration

=(k*Q*(-2*Q)/d)+(k*Q*(-2*Q)/d)+(k*Q*Q/(sqrt(2)*d))+(k*(-2*Q)*Q/d)+(k*(-2*Q)*(-2*Q)/(sqrt(2)*d))+(k*Q*(-2*Q)/d)

=(k*Q^2/d)*(-2-2+0.707-2+2.8284-2)=-4.4646*k*Q^2/d

problem 3:

part a:

let there be a square with coordinates:

(0,0) -> charge placed =Q
(d,0) -> charge placed=-Q
(d,d) -> charge placed=Q
(0,d) -> charge placed=-Q

then both potential and electric field will be zero.

electric potential energy=(k*Q*(-Q)/d)+(k*Q*(-Q)/d)+(k*Q*Q/(sqrt(2)*d))+(k*(-Q)*Q/d)+(k*(-Q)*(-Q)/(sqrt(2)*d))+(k*Q*(-Q)/d)

=(k*Q^2/d)*(-1-1+0.707-1+0.707-1)

=-2.586*k*Q^2/d

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