# Electric Potential **Electric potential** (often called potential) is a fundamental concept in electrostatics that describes the amount of work energy needed to move a unit of positive charge from a reference point to a specific point in an electric field. It is a scalar quantity that provides a way to characterize the energy of a location within an electric field, independent of the charge being placed there. ## Electric Potential Energy vs. Electric Potential These two concepts are closely related but distinct. - **Electric Potential Energy ($U_E$):** This is the potential energy a specific charge has due to its position in an electric field. It is the work done to move that charge from a reference point to its current location. - **Electric Potential ($V$):** This is a property of a location in an electric field. It is the electric potential energy per unit charge. **Relationship:** $$ V = \frac{U_E}{q} $$ By definition, it is also the work done ($W$) per unit charge: $$ V = \frac{W}{q} $$ ## Units of Electric Potential - The SI unit for electric potential is the **Volt (V)**. - One Volt is defined as one Joule of work per Coulomb of charge: $$ 1 \, \text{V} = 1 \, \text{J/C} $$ ## Calculation of Electric Potential ### a) For a Single Point Charge The electric potential ($V$) at a distance ($r$) from a single point source charge ($Q$) is given by: $$ V = k \frac{Q}{r} $$ - $k$ is Coulomb's constant ($k \approx 9 \times 10^9 \, \mathrm{N} \cdot \mathrm{m}^2 / \mathrm{C}^2$). - $Q$ is the source charge. - $r$ is the distance from the source charge. - **Note:** Unlike the electric field (a vector), potential is a **scalar**. The sign of the charge $Q$ is included in the calculation. A positive charge creates a positive potential, and a negative charge creates a negative potential. ### b) For Multiple Charges The total electric potential at a point due to a collection of charges is simply the **algebraic sum** of the potentials created by each individual charge. $$ V_{\text{total}} = k \sum_{i} \frac{q_i}{r_i} = k \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \ldots \right) $$ ## Potential Difference ($\Delta V$) In practice, we are often more interested in the **potential difference** (or voltage) between two points, as this is what drives the flow of charge (current). $$ \Delta V = V_B - V_A = \frac{W_{AB}}{q} $$ This is the work required per unit charge to move a charge from point A to point B. ## Equipotential Surfaces An equipotential surface is a surface on which the electric potential is the same at every point. No work is done in moving a charge between two points on an equipotential surface. ## The Electron-Volt (eV) In atomic and nuclear physics, a more convenient unit of energy is the **electron-volt (eV)**. - **Definition:** One electron-volt is the amount of energy gained or lost by a single electron when it moves across a potential difference of one volt. - **Conversion:** $1 \, \text{eV} = (1.602 \times 10^{-19} \, \text{C}) \times (1 \, \text{V}) = 1.602 \times 10^{-19} \, \text{J}$. ## Possible Questions/Answer - **Q:** What is the electric potential at a distance of 3 meters from a point charge of $5 \, \mu C$? **A:** $$ V = k \frac{Q}{r} = (9 \times 10^9) \frac{5 \times 10^{-6}}{3} = 3 \times 10^9 \times 5 \times 10^{-6} = 15 \times 10^3 \, \text{V} = 15 \, \text{kV} $$ - **Q:** What is the potential difference if it takes 600 J of work to move a 2 C charge between two points? **A:** $$ \Delta V = \frac{W}{q} = \frac{600 \, \text{J}}{2 \, \text{C}} = 300 \, \text{V} $$