# 19.1 ELECTRIC POTENTIAL ENERGY AND ELECTRIC POTENTIAL This section introduces the concepts of electric potential energy, electric potential, and potential difference. It explains how energy is stored in an electric field and how this relates to the work done on a charge. This is analogous to concepts found in . ## Electric Potential Energy ($U$) **Definition:** Electric potential energy is the energy required to move a charge against an electric field. It is also defined as the energy possessed by a charge due to its position in an electric field. **Factors:** The electric potential energy of an object depends on two main factors: 1. Its own electric charge ($q_0$). 2. Its position relative to other charged objects (i.e., its location within the electric field). **Change in Potential Energy:** The change in potential energy ($\Delta U$) when moving a charge $q_0$ from point B to point A is equal to the work done ($W_{BA}$) against the electric field. $$ \Delta U = U_A - U_B = W_{BA} $$ - $U_A$ and $U_B$ are the potential energies at points A and B, respectively. **Unit:** The SI unit for electric potential energy is the **joule (J)**. ## Electric Potential ($V$) and Potential Difference ($\Delta V$) **Electric Potential ($V$):** - This is a property of a *location* within an electric field, independent of the test charge placed there. - It is defined as the electric potential energy per unit charge. - For example, if a location has a potential of 12 J/C, a 2 C charge at that location would have 24 J of potential energy. **Potential Difference ($\Delta V$):** - **Definition:** The potential difference between two points is the work done in moving a unit positive charge from one point to another while maintaining electrostatic equilibrium. - **Formula:** The potential difference between points A and B is given by: $$ \Delta V = V_A - V_B = \frac{W_{AB}}{q_0} = \frac{\Delta U}{q_0} $$ - This can be rearranged to find the change in potential energy: $$ \Delta U = q_0 \Delta V $$ **Unit of Electric Potential:** - The SI unit is the **volt (V)**, which is equivalent to a **joule per coulomb (J/C)**. - **Definition of 1 Volt:** The potential difference between two points is one volt if one joule of work is done to move one coulomb of charge from one point to the other. $$ 1 \text{ volt} = \frac{1 \text{ joule}}{1 \text{ coulomb}} $$ - Common multiples and sub-multiples: | Prefix | Symbol | Value | | :--- | :--- | :--- | | millivolt | mV | $10^{-3}$ V | | microvolt | $\mu$V | $10^{-6}$ V | | kilovolt | kV | $10^{3}$ V | | megavolt | MV | $10^{6}$ V | | gigavolt | GV | $10^{9}$ V | ## Electric Field as Potential Gradient The relationship between electric field intensity ($E$) and potential difference is given by the potential gradient. $$ E = -\frac{\Delta V}{\Delta r} $$ The negative sign indicates that the direction of $E$ is along the decreasing potential. ## Possible Questions and Answers **Q:** A motorcycle battery and a car battery both have a voltage of 12.0 V, but the car battery can move a much larger charge (60,000 C vs. 5,000 C). Which one delivers more energy? **A:** The car battery delivers more energy. Energy is the product of charge and voltage ($\Delta U = q \Delta V$). Since the car battery can move more charge at the same voltage, it delivers more total energy ($7.20 \times 10^5$ J vs. $6.00 \times 10^4$ J). **Q:** What is the fundamental difference between electric potential and electric potential energy? **A:** Electric potential is a property of a *location* in space (energy per unit charge), whereas electric potential energy is the energy a *specific charge* has at that location. | Concept | Formula | Unit | | :--- | :--- | :--- | | Change in Potential Energy | $\Delta U = U_A - U_B = W_{BA}$ | Joule (J) | | Potential Difference | $\Delta V = \frac{\Delta U}{q_0}$ | Volt (V) | | Energy from Voltage | $\Delta U = q_0 \Delta V$ | Joule (J) |