## 19.5 ENERGY STORED IN A CAPACITOR A capacitor stores energy in the form of an **electric field**, which is a type of electrical potential energy. This energy is related to the charge $Q$ on the plates and the potential difference $V$ across them. ### Calculating Stored Energy When a capacitor is charged, its voltage increases from zero to a final voltage $V$. The charge $Q$ on the capacitor is directly proportional to the potential difference $V$ across it. This linear relationship is represented by a straight-line graph passing through the origin. The total electric potential energy $E$ stored in the capacitor is equal to the work done to charge it. This can be determined from the area under the potential-charge ($V$-$Q$) graph, which forms a right-angled triangle. - **Energy ($E$)** = Area of triangle OAB - $E = \frac{1}{2} \times \text{Base} \times \text{Height}$ - $E = \frac{1}{2} Q V$ ### Formulas for Stored Energy There are three equivalent formulas to calculate the energy stored in a capacitor, derived from the relationship $Q = CV$. | Formula | Variables | When to Use | | :--- | :--- | :--- | | $$E = \frac{1}{2} Q V \longrightarrow (19.11a)$$ | Charge ($Q$), Voltage ($V$) | When charge and voltage are known. | | $$E = \frac{1}{2} C V^{2} \longrightarrow (19.11b)$$ | Capacitance ($C$), Voltage ($V$) | When capacitance and voltage are known. | | $$E = \frac{Q^{2}}{2C} \longrightarrow (19.11c)$$ | Charge ($Q$), Capacitance ($C$) | When charge and capacitance are known. | ### Energy Stored in Electric Field and Energy Density The energy can also be expressed in terms of the electric field $E$ between the plates. For a parallel plate capacitor where $C = \frac{\epsilon_0 A}{d}$ and $V = Ed$: $$E = \frac{1}{2} (\frac{\epsilon_0 A}{d}) (Ed)^2 = \frac{1}{2} \epsilon_0 E^2 (Ad)$$ Since $(Ad)$ is the volume between the plates, we define **Energy Density** ($u$) as energy per unit volume: $$u = \frac{1}{2} \epsilon_0 E^2$$ ### Capacitor vs. Battery While both store energy, their charge and discharge characteristics differ significantly. - **Capacitor**: Stores energy in an electric field. It can be charged quickly and can release its entire stored energy almost instantaneously. - **Battery**: Stores energy chemically. It releases energy slowly through chemical reactions. ### Uses of Capacitors Capacitors are crucial components in many electronic circuits and devices: - **Filtering**: Removing unwanted frequencies in radios and TV sets. - **Energy Storage**: Storing and rapidly releasing energy in camera flashes. - **Motors**: Assisting in starting and running motors, such as in refrigerators and fans. - **Power Supplies**: Smoothing out rectified DC voltage by acting as a filter to reduce ripple. ### Possible Questions and Answers **Q:** A capacitor with a capacitance of 50 F is charged to a potential of 100 V. Calculate the energy stored in it. **A:** Using the formula $E = \frac{1}{2} C V^{2}$: $$E = \frac{1}{2} \times 50 \text{ F} \times (100 \text{ V})^{2}$$ $$E = \frac{1}{2} \times 50 \times 10000 = 250,000 \text{ J}$$ $$E = 250 \text{ kJ}$$ **Q:** Calculate the energy stored in a capacitor having 5 C of charge and a potential difference of 15 V. **A:** Using the formula $E = \frac{1}{2} Q V$: $$E = \frac{1}{2} \times 5 \text{ C} \times 15 \text{ V} = 37.5 \text{ J}$$