# 19.2 ELECTRIC POTENTIAL DUE TO A POINT CHARGE This section details the derivation of the electric potential created by a single, isolated point charge. Unlike a uniform electric field, the field from a point charge varies with distance, requiring a different approach for calculating potential. This concept is analogous to gravitational potential which you can review in . ## 1. Derivation Strategy The electric potential difference in a constant electric field $E$ is given by $\Delta V = -E \Delta r$. However, the electric field from a point charge $Q$ is non-uniform, varying with the square of the distance ($E \propto 1/r^2$). To handle this, the path of a test charge $q$ moving from a distant point $N$ to a point $1$ is divided into many small steps of length $\Delta r$. Over each tiny step, the electric field $E$ is assumed to be approximately constant. ## 2. Work Done Calculation The work done ($\Delta W$) in moving the charge $q$ through one small step from point $N$ to $N-1$ is calculated. The applied force is opposite to the electric field direction ($\theta = 180^{\circ}$). $$\Delta W_{N \rightarrow N-1} = -q E \Delta r$$ Substituting the formula for the electric field of a point charge, $E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}$, and using an approximation for the average distance ($r^2 \approx r_{N-1}r_N$), the work for one step simplifies to: $$\Delta W_{N \rightarrow N-1} = \frac{Qq}{4\pi\varepsilon_0}\left(\frac{1}{r_{N-1}} - \frac{1}{r_N}\right)$$ The total work done to move the charge from point $N$ to $1$ is the sum of the work done in all the individual steps. This results in a telescoping series where intermediate terms cancel out. $$\Delta W_{N \rightarrow 1} = \frac{Qq}{4\pi\varepsilon_0}\left(\frac{1}{r_1} - \frac{1}{r_N}\right)$$ ## 3. Electric Potential Energy (U) To define an absolute potential, the starting point $N$ is taken to be at infinity ($r_N = \infty$), where the potential is defined as zero. Since $1/\infty = 0$, the equation for total work simplifies. This work done in bringing the charge $q$ from infinity to a distance $r$ from $Q$ is defined as the **electric potential energy (U)** of the charge $q$ at that point. $$U = \frac{1}{4\pi\varepsilon_0}\frac{Qq}{r}$$ ## 4. Electric Potential (V) **Definition:** Electric potential $V$ at a point is the work done per unit positive charge in bringing it from infinity to that point. It is the potential energy per unit charge. $$V = \frac{U}{q}$$ Substituting the expression for $U$, we get the formula for the electric potential at a distance $r$ from a point charge $Q$. $$V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$$ This equation shows that the potential is the same for all points at an equal distance $r$ from the source charge $Q$. These points form an equipotential surface (a sphere in this case). | Formula | Description | Equation | | :--- | :--- | :--- | | **Electric Potential Energy** | The potential energy $U$ of a charge $q$ at a distance $r$ from a source charge $Q$. | $$U = \frac{1}{4\pi\varepsilon_0}\frac{Qq}{r}$$ | | **Electric Potential** | The potential $V$ at a distance $r$ from a source charge $Q$. | $$V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$$ |