Charging and Discharging of a Capacitor: MDCAT Physics notes

Capacitor and capacitance

A capacitor is a device that stores charge, and with it electrical potential energy in the electric field between its plates. The charge on it is proportional to the potential difference:

$$Q = CV$$

$C$ is the capacitance. SI unit: farad, 1 F = 1 C V$^{-1}$.

Parallel plate capacitor

$$C = \frac{\varepsilon_0\varepsilon_rA}{d}$$

• Increases with plate area $A$ and dielectric constant $\varepsilon_r$.
• Increases when separation $d$ decreases.
• Does not depend on the charge, the voltage or the thickness of the plates.

Examples: doubling the area and halving $d$ gives $4C$. Doubling the length, width and separation gives $(2\times2)/2 = 2C$.

Role of the dielectric

A dielectric increases $C$ by $\varepsilon_r$. For a capacitor that is isolated (charge fixed), inserting a dielectric reduces the field between the plates by $\varepsilon_r$; removing it makes the field rise again.

Energy stored

As the capacitor charges, the voltage rises from 0 to $V$, so the average voltage is $V/2$:

$$U = \tfrac12QV = \tfrac12CV^2 = \frac{Q^2}{2C}$$

The energy per unit volume is $u = \tfrac12\varepsilon_0\varepsilon_rE^2$.

• Doubling $Q$ (same $C$) gives $4U$.
• Doubling $E$ between the plates gives $4U$.
• Charged capacitor with energy $U$, disconnected, joined in parallel to an identical uncharged one: charge shares equally, each has $Q/2$, so each stores $U/4$ (total $U/2$; the rest is lost as heat and radiation).

Combinations

In parallel, the larger capacitor stores more energy ($\tfrac12CV^2$). In series, the same $Q$ means the smaller capacitor stores more energy ($Q^2/2C$).

Charging and discharging through a resistor

When a capacitor charges through a resistance $R$, charge builds up exponentially:

$$q = Q_0\left(1 - e^{-t/RC}\right) \qquad \text{(discharge: } q = Q_0e^{-t/RC})$$

• $RC$ is the time constant; ohm × farad = second.
• In one time constant the capacitor charges to about 63% of its final charge; on discharge it falls to about 37%.
• Theoretically full charge is reached only after infinite time; in practice about 5 time constants.
• Large $RC$ means slow charging and discharging.

Ultracapacitors

Ultracapacitors (supercapacitors) store charge using an electrical double layer at the electrode surface, giving very large capacitance.

Key formulas

• $Q = CV$; $C = \varepsilon_0\varepsilon_rA/d$
• $U = \tfrac12CV^2 = \tfrac12QV = Q^2/2C$
• Series: $1/C_e = \sum 1/C_i$; parallel: $C_e = \sum C_i$
• $\tau = RC$

Common MDCAT traps

• Energy is $\tfrac12QV$, not $QV$.
• Capacitance depends on geometry and medium, not on $Q$ or $V$.
• Parallel capacitors add directly; series use reciprocals (opposite of resistors).
• Energy goes as $Q^2$ or $E^2$: doubling gives four times.
• Plate thickness has no effect on $C$.

Quick revision

• 1 F = 1 C V$^{-1}$.
• $RC$ has the unit of time.
• Capacitor stores energy in its electric field.
• Closer plates mean larger capacitance.
• Ultracapacitors use the double-layer effect.