2.1. ELECTRIC OR ELECTROSTATIC POTENTIAL ENERGY
1. What is electric or electrostatic potential energy? Derive an expression for electric potential energy.
Electrostatic potential energy: Electric potential energy of a charge at a point in the electric field due to another charge is defined as the work done by an external force in bringing a test charge (without acceleration) from infinity to that point in the electric field.
Let a positive test charge $q_0$ be placed at A in an electric field $\vec{E}$ due to another positive charge Q (called source charge) as shown in figure 1. The force acting on the test charge $q_0$ in the electric field $\vec{E}$ is given by
$\vec{F}_e = q_0 \vec{E}$
This force tends to move the test charge in the direction of the electric field $\vec{E}$ (i.e., away from +Q charge).
Suppose an external force $\vec{F}_o$ acts just to overcome the electric force $\vec{F}_e$ on the test charge to move it without any acceleration towards source charge Q.
If the test charge is displaced through $d\vec{r}$ from point A to point B in the electric field $\vec{E}$, then the work done by the external force $\vec{F}_o$ is given by
$W_{AB} = \int_A^B \vec{F}_o \cdot d\vec{r}$
$\vec{F}_o = -\vec{F}_e = -q_0 \vec{E}$
$W_{AB} = \int_A^B \vec{F}_o \cdot d\vec{r}= -\int_A^B q_0 \vec{E} \cdot d\vec{r}$
$W= -q_0 \int_A^B \vec{E} \cdot d\vec{r}$
This work done gives the difference in electric or electrostatic potential energy of the test charge between points A and B in the electric field.
$W_{AB} = U_B - U_A = -q_0 \int_A^B \vec{E} \cdot d\vec{r}$
Thus, difference in electric potential energy between two points A and B in the electric field is the work done in moving a test charge without acceleration from point A to point B in the electric field of the source charge.
If point A is at infinity, then, there is no electrostatic force between charge Q and the test charge $q_0$. Thus, the potential energy of the test charge $q_0$ at infinity is zero. The work done to displace the test charge (without acceleration) from infinity to the point B in the electric field is given by,
$W_{\infty B} = U_B - U_{\infty} = U_B - 0 = U_B$
or
$U_B = W_{\infty B} = -q_0 \int_{\infty}^B \vec{E} \cdot d\vec{r}$
This work done is equal to the electrostatic potential energy of the test charge at any point (say B) in the electric field.
