Electrostatics MCQ

The electric field and the electric potential at a point are E and V.
a. If E=0, V must be 0
b. If V=0, E must be 0.
c. if E !=0, V cannot be 0.
d. if V !=0, E cannot be 0.

Ans: none

Please give reason

for the first case
1. E= 0 V must be zero
this not obvious well E is defined as –dV/dr so if E is zero that means that V is constant, but is not zero.
2.V= 0 then E must be zero. Now for this case consider a dipole along x axis in the xy plane then, at any point on the perpendicular bisector of the line joining the charges, the electric potential is zero. This can be seen from a direct computation. But the electric field along the perpendicular bisector varies as 1/x^3. SO this case is also false.

Well coming to cases 3 and 4 I don’t understand what they mean (symbols not clear). But however they still depend some reasoning

Hope this helps
Pinnaka

!= means not equal to

of if E # 0, V cannot be zero. Once again this statement is false. Now consider a dipole configuration. Along the axis of the dipole and between the charges, the electric field is never zero but V is zero exactly at the midpoint of the lne segement joining the two charges of the dipole. therefore V can be zero

if V # 0 then E cannot be zero
once again the statement is false, consider a two + charges seperated by certan distance. The V is not zero at any point along the line segement joining the charges but E is zero at the midpoint of the line segement joining the charges.

Therefore all cases are false

Hoipe this helps
Pinnaka