Question

The electric potential V at any point P(x,y,z) in space is given by V=$4{x^2}$ V. The electric field at the point (1m,0m,2m) in volt/meter is:

A. 8 along negative x axis
B. 8 along positive x axis
C. 16 along negative x axis
D. 16 along positive x axis

Answer 1

Hint: When we do any work against the conservative force then the work done by us will be stored in the form of potential energy in the system and potential energy increases. Along the direction of conservative force then potential energy of the system decreases.

Formula used:
${F_c} = - \dfrac{{dU}}{{dx}}$

Complete step-by-step answer:
Let us assume there is an object. We are displacing that object up very slowly which means at every instant we can assume its velocity will be zero. When we are moving an object upwards which means that we are displacing the object against the gravitational force. That literally means we are doing some work and according to conservation of energy that work will not go in vain. It will get converted in some form and that is nothing but in the form of potential energy.
If we clearly observe the above case, as the object is moving against gravity i.e as the work done by gravity is negative, the potential energy of the system is increasing.
Hence from the above relation we have the formula ${F_c} = - \dfrac{{dU}}{{dx}}$
${F_c}$ is conservative force and ‘U’ is the potential energy and negative sign indicates that along the direction of conservative force, potential energy of the system decreases.
Similarly, the electrostatic force is also conservative force.
$\eqalign{
  & {F_c} = Eq \cr
  & \Rightarrow U = qV \cr
  & \Rightarrow {F_c} = - \dfrac{{dU}}{{dx}} \cr
  & \Rightarrow Eq = - \dfrac{{d\left( {qV} \right)}}{{dx}} \cr
  & \Rightarrow E = - \dfrac{{dV}}{{dx}} \cr
  & \Rightarrow E = - \dfrac{{d\left( {4{x^2}} \right)}}{{dx}} \cr
  & \Rightarrow E = - 8x\mathop i\limits^ \wedge \cr
  & \therefore E = - 8\mathop i\limits^ \wedge {\text{ at x = 1m}} \cr} $

So, the correct answer is “Option A”.

Note: Potential energy is valid only if conservative forces are present. Gravity, spring force and electrostatic force are the examples for conservative forces. In the question it doesn’t matter which coordinates they give, the electric field depends only upon the x coordinate as we had got that expression.