Capacitors and Capacitance: Principles, Formulas, and Circuit Rules

Capacitors & Capacitance

Introduction to Capacitors

A capacitor consists of two conductors separated in space, carrying equal and opposite charges. The electric field between the conductors stores electrostatic energy, and a potential difference is established between them.

• Capacitor: Device that stores electric charge and energy in the electric field between its plates.

• Potential Difference (Vab): The voltage between the two conductors is proportional to the charge Q on the plates.

• Capacitance (C): The proportionality constant that relates charge and potential difference. Defined as:

$C = \frac{Q}{V_{ab}}$

• Unit: Farad (F)

Example: If a capacitor has a charge of 1 C and a potential difference of 1 V, its capacitance is 1 F.

Parallel Plate Capacitors

Structure and Electric Field

A parallel plate capacitor consists of two parallel metal plates of area A, separated by a distance d. The electric field between the plates is uniform and given by:

$E = \frac{Q}{\varepsilon_0 A}$

• Q: Charge on each plate

• \(\varepsilon_0\): Permittivity of free space (vacuum)

• A: Area of each plate

Potential Difference:

$V = Ed = \frac{Qd}{\varepsilon_0 A}$

Capacitance of a Parallel Plate Capacitor:

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

• Capacitance depends only on the geometry (area and separation) of the plates and the permittivity of the medium between them.

The Permittivity of Free Space (\(\varepsilon_0\))

• \(\varepsilon_0\): A fundamental physical constant describing the ability of vacuum to permit electric field lines.

• Value: $\varepsilon_0 = 8.854 \times 10^{-12} \ \frac{C^2}{N \cdot m^2}$

• Appears in Coulomb's law, often hidden in the constant k:

$k = \frac{1}{4\pi \varepsilon_0}$

Behavior of Plate Capacitors

Changing Plate Separation

• If the plates are moved closer together (with fixed potential difference), the capacitance increases.

• If the plates are moved apart (with fixed charge), the potential difference increases and capacitance decreases.

Key Questions:

• At fixed voltage, decreasing d increases C and increases Q.

• At fixed charge, decreasing d keeps Q constant, but V decreases.

Capacitor Circuit Diagrams and Rules

Basic Circuit Representation

• Wires are treated as resistance-less and as equipotentials.

• Potential changes occur only across components (e.g., batteries, capacitors).

• A battery maintains a fixed potential difference between two points in a circuit.

Charge Conservation in Circuits

• Charge does not accumulate in a circuit; the total charge remains zero.

• When a capacitor is charged, positive charge builds up on one plate and negative charge on the other.

• In a closed circuit, the movement of charge completes the loop, ensuring conservation.

Capacitors in Series

Series Combination

When capacitors are connected in series, the total (equivalent) capacitance is less than any individual capacitance in the series.

• The same charge Q flows through each capacitor.

• The total voltage is the sum of voltages across each capacitor.

Formula for Series Capacitance:

$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$

• This is not the same as $C = C_1 + C_2$.

• For two identical capacitors: $C = \frac{C_1}{2}$

Example: If $C_1 = 1\ F$ and $C_2 = 1\ F$, then $C = 0.5\ F$.

Summary Table: Series vs. Parallel Capacitors

Configuration	Charge (Q)	Voltage (V)	Equivalent Capacitance (Ceq)
Series	Same on all capacitors	Adds: $V = V_1 + V_2 + ...$	$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$
Parallel	Adds: $Q = Q_1 + Q_2 + ...$	Same across all capacitors	$C_{eq} = C_1 + C_2 + ...$

Additional info: The notes also include conceptual questions and circuit diagrams to reinforce understanding of how charge and voltage behave in different capacitor configurations.