Capacitors and Capacitance: Principles, Formulas, and Circuit Applications

Capacitors & Capacitance

Introduction to Capacitors

Capacitors are devices formed by two conductors separated in space, carrying equal and opposite charges. They store electrostatic energy in the electric fields between the conductors.

• Capacitor: An arrangement of two conductors with electric fields filling the space between them.

• Potential Difference: A voltage is set up between the conductors, proportional to the charge $Q$.

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

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

• Units: Farads (F)

Electric Field and Capacitance

The electric field between the plates of a capacitor is uniform and can be calculated as:

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

• $\varepsilon_0$: The permittivity of free space, a fundamental constant describing the strength of electric fields in vacuum.

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

• Coulomb's Law Constant ($k$): $k = \frac{1}{4\pi \varepsilon_0}$

Parallel Plate Capacitors

Geometry and Capacitance

Parallel plate capacitors consist of two metal plates of area $A$ separated by a distance $d$. The capacitance depends only on the geometry:

• Electric Field: $E = \frac{Q}{\varepsilon_0 A}$

• Potential Difference: $V = Ed = \frac{Qd}{\varepsilon_0 A}$

• Capacitance: $C = \frac{\varepsilon_0 A}{d}$

Effects of Plate Separation

• If the plates are moved closer together (decreasing $d$), capacitance increases.

• If the potential difference is held constant and plates are moved closer, charge on each plate increases.

• If the capacitor is disconnected from the battery and plates are moved, charge stays the same, but potential difference decreases.

Capacitor Circuit Diagrams and Rules

Basic Circuit Principles

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

• Potential changes occur only when electrical components are attached.

• A battery sets a fixed potential difference across its terminals.

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

Charge Movement in Circuits

• Positive charge builds up on one plate, pushing positive charge off the other plate.

• In a closed circuit, 'pushed-off' charges move around the circuit, forming the first set of positive charges.

Capacitors in Series

Series Connection Principles

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

• Formula for Series Capacitance:

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

• Key Point: The formula is NOT $C = C_1 + C_2$.

• For two identical capacitors in series ($C_1 = C_2 = 1$ F): $C = \frac{1}{2}$ F

Example Calculation

• If $C_1 = 1$ F and $C_2 = 1$ F:

$\frac{1}{C} = \frac{1}{1} + \frac{1}{1} = 2$

$C = \frac{1}{2}$ F

Summary Table: Series vs. Parallel Capacitors

Connection Type	Formula for Equivalent Capacitance	Key Property
Series	$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + ...$	Charge is the same on all capacitors
Parallel	$C = C_1 + C_2 + ...$	Voltage is the same across all capacitors

Additional info:

• These notes cover the foundational concepts of capacitors, including their physical principles, mathematical relationships, and behavior in electric circuits. The content is suitable for introductory college-level physics courses.