Question

An electric field is expressed as $E=2i+3j$. Find the potential difference between two points A and B whose position vectors are given by ${{r}_{a}}=i+2j,{{r}_{b}}=2i+j+3k$
A) -1V
B)1V
C)2V
D)3V

Answer 1

Hint: Let us find the distance vector between the given two points a and b in the above question. Next, we need to find the relation between potential difference and electric field. The potential difference between two points will be equal to the dot product of the electric field and the distance vector.
Formulas used:
$P.E=E.S$

Complete answer:
Let us first write down the given terms,
$\begin{align}
  & E=2i+3j \\
 & \Rightarrow {{r}_{a}}=i+2j \\
 & \Rightarrow {{r}_{b}}=2i+j+3k \\
\end{align}$
Now, let us find the distance vector between the two points a and b.
The distance vector is equal to,
$\begin{align}
  & {{r}_{b}}-{{r}_{a}}=(2i+j+3k)-(i+2j) \\
 & \Rightarrow {{r}_{b}}-{{r}_{a}}=i-j+3k \\
\end{align}$
Therefore, the potential difference between the points a and b is,
$\begin{align}
  & P.E=(2i+3j).(i-j+3k) \\
 & \Rightarrow P.E=2-3+0 \\
 & \Rightarrow P.E=-1V \\
\end{align}$

Therefore, the correct option is option A.

Additional information:
Electric potential is a quantity that expresses the amount of potential energy per unit of charge at a specified location. When a Coulomb of charge (or any given amount of charge) possesses a relatively large quantity of potential energy at a given location, then that location is said to be a location of high electric potential. And similarly, if a Coulomb of charge (or any given amount of charge) possesses a relatively small quantity of potential energy at a given location, then that location is said to be a location of low electric potential. As we begin to apply our concepts of potential energy and electric potential to circuits, we will begin to refer to the difference in electric potential between two points. When a force does work on an object, potential energy can be stored. An object with potential energy has the potential to do work. (It's not doing work right now, but it has the potential.) An object has potential energy by virtue of its position. Work and potential energy are closely related. Additional potential energy stored in an object is equal to the work done to bring the object to its new position.

Note:
The potential difference between two points can be calculated by solving the dot product of the distance vector and the electric field. The potential difference is a scalar quantity. Dot product of two vector quantities is a scalar quantity.