Electric Field and Electric Potential

E and V: Definitions
Electric field (E) is a vector quantity that describes the force experienced by a positive test charge placed at a point in space. It is defined as the force per unit positive charge.

Electric potential (V) is a scalar quantity that measures the electric potential energy per unit positive charge at a point in space. It represents the work done in bringing a positive test charge from infinity to that point.

a. E=0 implies V=0?
If the electric field (E) at a point is zero, it means that there is no force acting on a positive test charge placed at that point. However, this does not necessarily imply that the electric potential (V) at that point is zero.

Even when the electric field is zero, it is possible to have a non-zero electric potential. This can occur when the point in question is at a certain distance from a charged object or in the presence of a non-uniform electric field. In such cases, the electric potential may be influenced by the surrounding charges or the shape of the electric field.

Therefore, it is not always true that if E is zero, V must also be zero.

b. V=0 implies E=0?
If the electric potential (V) at a point is zero, it means that the electric potential energy per unit positive charge is zero. This implies that there is no potential difference between that point and a reference point (often taken as infinity).

In the absence of a potential difference, there will be no force acting on a positive test charge. Hence, if V is zero, it can be concluded that the electric field (E) at that point is also zero.

Therefore, if V is zero, E must also be zero.

c. E ≠ 0 implies V ≠ 0?
If the electric field (E) at a point is not equal to zero, it means that there is a force acting on a positive test charge placed at that point. In this case, the electric field is responsible for the motion of the charge.

The electric potential (V) at a point, however, is influenced by the presence of charges in the surrounding area. It represents the potential energy per unit positive charge and is not solely dependent on the electric field.

Therefore, if E is not zero, it does not necessarily mean that V cannot be zero. The electric potential can still be zero if the charges in the vicinity cancel out each other's contributions to the potential.

d. V ≠ 0 implies E ≠ 0?
If the electric potential (V) at a point is not equal to zero, it means that there is a non-zero potential difference between that point and a reference point. This potential difference gives rise to an electric field (E) in the region.

The electric field is responsible for the force experienced by a positive test charge. Hence, if the potential is non-zero, it implies that there is an electric field present.

Therefore, if V is not zero, it implies that E cannot be zero.

Conclusion
In summary:
a. E=0 does not imply V=0.
b. V=0 implies E=0.
c. E ≠ 0 does not imply V cannot be zero.
d. V ≠ 0 implies E cannot be zero.

It is important to note that the relationships

B