Potential Energy in an External Field

Potential Energy in an External Field

When a charge is placed in an external electric field, it experiences a force. To bring the charge from infinity (where the field is zero) to a specific point in the field, work must be done against this force. This work gets stored as potential energy of the charge in the external field.

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1. Potential Energy of a Single Charge $q$ in an External Field

Consider a point charge $q$ placed in an external electric field $\mathbf{E}$ (produced by other charges not included in the system). The electrostatic potential at a point in this field is defined as the work done in bringing a unit positive charge from infinity to that point.

• If the potential at a point is $V$, then the potential energy $U$ of a charge $q$ placed there is:

• Meaning: $U$ is the work done by an external agent in bringing the charge $q$ from infinity to that point, without accelerating it.
• Sign convention: If $q$ and $V$ have the same sign, $U$ is positive (work is done against the field). If opposite signs, $U$ is negative (work is done by the field).

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2. Potential Energy of a System of Two Charges in an External Field

Now consider two point charges $q_1$ and $q_2$ placed at positions $\mathbf{r}_1$ and $\mathbf{r}_2$ in an external field $\mathbf{E}$. The total potential energy of the system is the sum of:

1. The work done to bring $q_1$ from infinity to $\mathbf{r}_1$ in the external field.
2. The work done to bring $q_2$ from infinity to $\mathbf{r}_2$ in the combined field of the external source and $q_1$.

Step-by-step derivation:

• Step 1: Bring $q_1$ from infinity to $\mathbf{r}_1$. The work done is:

where $V(\mathbf{r}_1)$ is the external potential at $\mathbf{r}_1$.

• Step 2: Now bring $q_2$ from infinity to $\mathbf{r}_2$. The external agent must work against:

  • The external field (potential $V(\mathbf{r}_2)$).
  • The field of $q_1$ (potential at $\mathbf{r}_2$ due to $q_1$ is $\frac{1}{4\pi\epsilon_0} \frac{q_1}{r_{12}}$, where $r_{12} = |\mathbf{r}_2 - \mathbf{r}_1|$).

  The work done in this step is:

• Total potential energy $U$ of the two-charge system in the external field is:

• Key insight: The term $\frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$ is the mutual potential energy of the two charges — it does not depend on the external field. It is the work done to assemble the two charges in the absence of any external field.

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3. Generalization to $n$ Charges …