# The electric field and electric potential at a point are E and V respectively: a). If E=0, V must...

## Question:

The electric field and electric potential at a point are E and V respectively:

a). If E=0, V must be 0.

b). If V=0, E must be zero.

c). If E is not equal to 0, V cannot be 0.

d). If V is not equal to 0, E cannot be 0.

## Electric Field and Electric Potential:

Consider a point charge Q. It will set up a conservative force field in the region of space around it. The force field will fall off in strength as the inverse square of the separation from the charge. The force field is also directed radially outward from the charge. These two features demand that mathematically the force field should be curl-free. Since the curl of the gradient of a scalar field is always zero we have an alternative description in terms of a scalar field. This scalar field is usually called the electrostatic potential.

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The strength of the electric field at a point is the force experienced by a unit test charge placed at that point. On the other hand, the electrostatic potential at a point is the work that must be done to bring in a unit positive charge from infinity to that point without accelerating it.

Electric field strength and the electrostatic potential are related as follows:

{eq}\displaystyle {\vec{E}=-\triangledown V} {/eq}

Electric field strength depends only on the change in potential and not the potential per se. Even if the potential at a point is zero then the electric field strength at that point need not be zero. For instance, consider an electric dipole. The potential at any point in its equatorial plane is zero. However, the electrostatic field is non zero. Similarly, even if the electric field at a point is zero then electrostatic potential need not be zero. For instance, inside a charged spherical conductor the electric field is zero but the electrostatic potential is non zero and equal to the surface value.

Even if the electric field strength at a point is non-zero the potential can be zero as we have seen in the case of the dipole. Also, even if the potential is non-zero the electric field can be zero as in the case of the interior of the conducting charged sphere.

Mathematically, potential obeys the Poisson's equation. This is a second-order equation. Hence the boundary conditions on the potential and its first derivative(which is nothing but the electric field strength) are independent. Therefore by changing the boundary conditions, any value can be chosen for one quantity irrespective of the value of the other.

Thus all the four options are FALSE.