A capacitor is a device that stores electrical energy in an electric field created between a pair of conductors (called 'plates'). The energy stored is equal to the work done to charge the capacitor. When a voltage source is connected across a capacitor, it moves charge from one plate to the other. This process of charging involves doing work against the electric field that builds up between the plates. This work is stored as electrical potential energy in the capacitor.
The energy stored in a capacitor is a form of electrical potential energy. It is stored within the electric field created between the capacitor's plates and is equivalent to the work done to charge the capacitor.
| Property | Details |
|---|---|
| Nature | Scalar. Energy has magnitude but no direction. |
| SI Units | Joule (J) |
| Other Common Units | Electron-volt (eV), especially for microscopic systems. |
| Dimensional Formula | M L^2 T^-2 |
| Conservation | In an ideal, isolated circuit (like an LC circuit), the total energy is conserved, oscillating between the capacitor's electric field and the inductor's magnetic field. In a charging RC circuit, only half the energy supplied by the battery is stored; the other half is dissipated as heat in the resistor. |
| Key Dependencies | The stored energy is directly proportional to the capacitance and to the square of the voltage or the square of the charge. |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| U | Potential Energy | Joule (J) | The electrical potential energy stored in the capacitor. |
| C | Capacitance | Farad (F) | A measure of the capacitor's ability to store charge. |
| V | Voltage | Volt (V) | The potential difference across the capacitor's plates. |
| Q | Charge | Coulomb (C) | The magnitude of the charge stored on each plate. |
The energy stored in a capacitor is equal to the work done to charge it. Consider the process of moving an infinitesimal amount of charge \(dq\) from the negative plate to the positive plate. The work done \(dW\) requires moving this charge against the existing potential difference \(V\) across the plates.
Using the definition of capacitance, \(C = q/V\), we can express the voltage \(V\) in terms of the charge \(q\) already on the plates: \(V = q/C\). Substituting this into the work equation gives:
To find the total work done \(W\) in charging the capacitor from zero charge to a final charge \(Q\), we integrate this expression:
Integrating this expression gives the total work done, which is the potential energy \(U\) stored in the capacitor:
By substituting the relationship \(Q = CV\) into this result, we can derive the other common forms of the energy equation.
The formula for the energy stored in a capacitor can be expressed in several equivalent forms, depending on which variables (Charge, Voltage, or Capacitance) are known. Another form describes the energy in terms of the electric field itself.
| Formula Variant | Description | When to Use |
|---|---|---|
| Using Capacitance and Voltage | U = (1/2) * C * V^2. This is the most commonly used form. | When the capacitance (C) and the voltage (V) across the capacitor are known. |
| Using Charge and Voltage | U = (1/2) * Q * V. This form directly relates to the work done moving charge across a potential difference. | When the total charge stored (Q) and the voltage (V) across the capacitor are known. |
| Using Charge and Capacitance | U = Q^2 / (2C). This form is derived by substituting V = Q/C. | When the total charge stored (Q) and the capacitance (C) are known, particularly useful when analyzing charge transfer between capacitors. |
| Using Electric Field (Energy Density) | U = (1/2) * ε * E^2 * (Volume). This expresses energy as a property of the electric field itself. | When analyzing the energy density (energy per unit volume) stored in the electric field (E) within the dielectric material (with permittivity ε). |
Energy Storage: Capacitors are fundamental components in power supplies for smoothing rectified DC voltage, providing a stable power source for electronic components.
Camera Flashes: They store a large amount of energy from a battery over several seconds and then release it very quickly (in milliseconds) to power the bright flash tube.
Defibrillators: Medical defibrillators use a large capacitor to store a high-energy charge, which can be delivered to a patient's heart to restore a normal rhythm during cardiac arrest.
Tuning Circuits: In radios and other communication devices, variable capacitors are used in resonant circuits to tune to specific frequencies, allowing you to select a radio station.
Electronic Power Supplies: In your phone charger or computer power supply, capacitors store energy to smooth out fluctuations from the AC wall outlet after it's converted to DC. This provides a stable voltage essential for sensitive electronics.
Car Audio Systems: High-power car stereos use large capacitors to meet sudden power demands. When a deep bass note hits, the amplifier needs a burst of current that the car's alternator can't supply instantly. The capacitor provides this quick energy boost, preventing lights from dimming and ensuring clean sound.
Utility Power Grids: Large banks of capacitors are used in power distribution systems for 'power factor correction'. They store and release energy to compensate for inductive loads (like large motors), which improves the overall efficiency and stability of the electrical grid.
| Quantity | Symbol | SI Unit | Dimensional Formula |
|---|---|---|---|
| Energy | U | Joule (J) | [M][L]²[T]⁻² |
| Capacitance | C | Farad (F = C/V) | [M]⁻¹[L]⁻²[T]⁴[I]² |
| Voltage | V | Volt (V = J/C) | [M][L]²[T]⁻³[I]⁻¹ |
| Charge | Q | Coulomb (C = A·s) | [I][T] |
A dimensional analysis of the formula \(U = \frac{1}{2}CV^2\) confirms its validity. The dimensions of the right side are: \([C][V]^2 = ([M]^{-1}[L]^{-2}[T]^4[I]^2) \cdot ([M][L]^2[T]^{-3}[I]^{-1})^2 = ([M]^{-1}[L]^{-2}[T]^4[I]^2) \cdot ([M]^2[L]^4[T]^{-6}[I]^{-2}) = [M][L]^2[T]^{-2}\), which are the dimensions of Energy.
The energy (U) stored in a capacitor can be calculated using three equivalent formulas: U = ½CV², U = Q²/(2C), or U = ½QV. These formulas determine the amount of electrical potential energy, measured in Joules (J), that is stored in the electric field between the capacitor's plates.
In these formulas, U represents the stored potential energy in Joules (J). C is the capacitance of the capacitor in Farads (F). V is the voltage or potential difference across the capacitor plates in Volts (V). Q is the magnitude of the charge stored on each plate in Coulombs (C).
You should choose the formula based on the two variables you are given in a problem. If you know capacitance (C) and voltage (V), use U = ½CV². If you know charge (Q) and capacitance (C), use U = Q²/(2C). If you are given charge (Q) and voltage (V), use U = ½QV.
A very frequent mistake is failing to convert the capacitance into the base unit of Farads (F) before calculating. Capacitance is often given in microfarads (µF), nanofarads (nF), or picofarads (pF), which must be converted to Farads (e.g., 1 µF = 1 x 10⁻⁶ F) to get the energy in Joules.
A camera flash is a perfect example. A capacitor slowly stores energy from the camera's battery and then releases this energy almost instantly to power the flash tube, creating a very bright, short burst of light. Medical defibrillators work similarly, storing a large amount of energy to deliver a life-saving electrical shock.
The energy stored in a capacitor is precisely equal to the work done to charge it. As a voltage source moves charge from one plate to the other, it does work against the established electric field. This work is converted into potential energy, which is then stored within that electric field.