Question

The plot of the x component of the electric field as a function of x in a certain region is shown in the figure. The y and z components of the electric field are zero in this region. The electrical potential at origin is 10V. The electric potential at $x = 2$ is

A. 10V
B. 20V
C. 30V
D. 40V

Answer 1

In this question initially a particle is at rest with a potential energy of 10V, now the particle is moved from the rest to the position$x = 2$, then find the potential energy of the particle at $x = 2$by using the area under the curve. For this, we need to find the area under the curve for the change in electric potential from origin to$x = 2$.

Complete step by step answer:
Since the y and z components of the electric field are zero hence, we can say the initial position of the particle is at $x = 0$where the electric potential is 10V,
Now the particle is being moved along the x-axis to the position $x = 2$as shown in the plot and also the potential energy changes.
In the given ${E_x} - x$curve, when the particle moves from position $x = 0$to$x = 2$, the potential curve moves in a negative direction hence we can write
Hence the electric potential of the particle at $x = 2$will be $= 30V$

$${V_B} - {V_A} = - \int {{E_x}dx} \\\ \Rightarrow {V_B} - 10 = - \left( {\dfrac{1}{2} \times 2 \times \left( { - 20} \right)} \right) \\\ \Rightarrow {V_B} - 10 = 20 \\\ \Rightarrow {V_B} = 30V \\\$$

So, the correct answer is “Option C”.

Note:
Electric potential is the amount of work needed to move a unit of electric energy from a reference point to a specific point in the electric field without acceleration being produced. Students must note that the potential difference between two points represents the work done in transferring a unit quantity from one point to the other.