📂Electrodynamics

Properties of Potential

Reference Point of Potential¹

The definition of potential is as follows.

$$ V(\mathbf{r} ) \equiv - \int _\mathcal{O} ^{\mathbf{r}} \mathbf{E} \cdot d \mathbf{l} $$

Therefore, depending on the reference point $\mathcal{O}$, its value can vary. For instance, if a new reference point $\mathcal{O}^{\prime}$ is chosen, there will be a difference in value by a certain constant $K$.

$$ \begin{align*} V^{\prime} (\mathbf{r} ) =&\ -\int _{\mathcal{O}^{\prime}}^\mathbf{r} \mathbf{E} \cdot d\mathbf{l} \\ =&\ -\int _{\mathcal{O}^{\prime}} ^\mathcal{O} \mathbf{E} \cdot d\mathbf{l} -\int_\mathcal{O} ^\mathbf{r} \mathbf{E} \cdot d \mathbf{l} \\ =&\ K + V( \mathbf{r} ) \end{align*} $$

$K$ is the value obtained by line integrating the electric field $\mathbf{E}$ from $\mathcal{O}$ to $\mathcal{O}^{\prime}$. Here, it’s crucial to note that although the value of potential $V$ can change due to the reference point, the value of the electric field $\mathbf{E}$ remains unchanged.

Since the derivative of a constant is $0$,

$$ \nabla V^{\prime} = \nabla (K+V)=\nabla K + \nabla V = \nabla V $$

Therefore,

$$ \nabla V^{\prime} =\nabla V =\mathbf{E} $$

Since choosing the reference point has no effect on the electric field, it provides a significant advantage when dealing with potential. What’s important is not the potential itself but the potential difference between two points. It can be shown that the potential difference is also independent of how the reference point is chosen.

$$ V^{\prime}( \mathbf{b} ) -V^{\prime}( \mathbf{a} ) = \left[ K + V( \mathbf{b} ) \right] - \left[ K +V( \mathbf{a} ) \right]= V(\mathbf{b}) - V( \mathbf{a}) $$

Principle of Superposition

Like the electric field, potential also follows the principle of superposition. The total potential is simply the sum of the potentials created by each source charge. If $V=V_{1}+V_2+V_{3}+\cdots$ and $\mathbf{E}_{i}=-\nabla V_{i}$,

$$ \begin{align*} \mathbf{E} =&\ \mathbf{E}_{1}+\mathbf{E}_2+\mathbf{E}_{3}+\cdots \\ =&\ -\nabla V_{1} -\nabla V_2 -\nabla V_{3} -\cdots \\ =&\ -\nabla (V_{1}+V_2+V_{3}+\cdots) \\ =&\ -\nabla V \end{align*} $$