Electric Potential Energy

Definition

Electric potential energy at a point is the work done against the electric field when a unit positive Charge is brought from infinity to that point. The unit charge is moved without accelerating.

Explanation

Consider an isolated charge $Q$ fixed in space. A test charge $q$ is initially at infinity and is moved towards the charge $+Q$. Due to the repulsive force between the two charges, external work must be done to move the test charge from infinity to a point near the charge $Q$.

• External work is required to bring the charge from infinity to point B due to the repulsion between the charges. This results in a negative work value.
• Let the test charge be at a large distance $r_A$ from charge $Q$. The distance between $r_A$ and $r_B$ is divided into infinitesimally small displacements $r_1, r_2, \dots, r_N$, ensuring that the field intensity remains constant over each displacement.

Work from $r_A$ to $r_1$

The test charge is moved from $r_A$ to $r_1$ with a small displacement $\Delta r$. Since external work is done to move the charge against the electric field, we include a negative sign in the equation.

$$\Delta W_{r_A \rightarrow r_1} = -F \Delta r \cos \theta$$

Substitute the expression for force $F = \frac{1}{4 \pi \varepsilon_0} \frac{Qq}{r^2}$:

$$\Delta W_{r_A \rightarrow r_1} = -\frac{1}{4 \pi \varepsilon_0} \frac{Qq}{r^2} \Delta r \cos \theta$$

Since $\theta = 180^\circ$ (because the test charge is moved against the electric field), we have:

$$\Delta W_{r_A \rightarrow r_1} = \frac{Q q}{4 \pi \varepsilon_0} \frac{1}{r^2} \Delta r$$

Rewriting in terms of $r_A$ and $r_1$:

$$\Delta W_{r_A \rightarrow r_1} = \frac{Qq}{4 \pi \varepsilon_0} \left( \frac{1}{r_A} - \frac{1}{r_1} \right)$$

Work from $r_1$ to $r_2$

Similarly, the work done in moving the test charge from $r_1$ to $r_2$ is given by:

$$\Delta W_{r_1 \rightarrow r_2} = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r_2} - \frac{1}{r_1} \right)$$

Work from $r_N$ to $r_B$

For the final displacement, the work done to move the charge from $r_N$ to $r_B$ is:

$$\Delta W_{r_N \rightarrow r_B} = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r_B} - \frac{1}{r_N} \right)$$

Total Work from $r_A$ to $r_B$

The total work done from $r_A$ to $r_B$ can be written as the sum of all the infinitesimal works:

$$\Delta W_{r_A \rightarrow r_B} = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r_1} - \frac{1}{r_A} + \frac{1}{r_2} - \frac{1}{r_1} + \dots + \frac{1}{r_B} - \frac{1}{r_N} \right)$$

Simplifying the series, we get:

$$\Delta W_{r_A \rightarrow r_B} = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r_B} - \frac{1}{r_A} \right)$$

The work done to move a test charge $q$ from infinity to a distance $r$ from $Q$ is:

$$\Delta W_{r_A \rightarrow r_B} = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r_B} - 0 \right) = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r_B} \right)$$

Thus, the work done is given by:

$$\Delta W = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)$$

Therefore, the electric potential energy $U$ is:

$$U = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)$$

Electric Potential

The electric potential at a distance $r$ from $Q$ is defined as:

$$V = \frac{W}{q}$$

Substituting $W = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)$:

$$V = \frac{\frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)}{q}$$

Simplifying:

$$V = \frac{Q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)$$

Thus, the electric potential is:

$$V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}$$

Summary

1. Electric Potential Energy: The work done to move a test Charge from infinity to a point near a charge $Q$ in an Electric Field.

2. Electric Potential: The potential energy per unit charge at a point in the field.

Important Relationships:

1. Work Done: $\Delta W = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)$
2. Electric Potential: $V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}$

Concept	Formula	Description
Electric Potential Energy	$U = \frac{Q q}{4 \pi \varepsilon_0} \left( \frac{1}{r} \right)$	Work done to move a test charge in an electric field.
Electric Potential	$V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}$	The potential energy per unit charge at a point in the field.

Electric potential energy and potential are crucial concepts in understanding how charges interact within electric fields. These principles have broad applications in areas such as electrostatics, capacitors, and more.

References

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