Capacitors, Energy Storage, and Dielectrics

Capacitors and Capacitance

Introduction to Capacitors

Capacitors are fundamental components in electrical circuits, used for storing energy and charge. They are widely utilized in timing devices, energy regulation, and rapid charging/discharging applications.

• Energy & Charge Storage: Capacitors can store electrical energy and release it quickly when needed.

• Rapid Charging/Discharging: Unlike batteries, capacitors can charge and discharge rapidly, making them suitable for applications requiring quick energy delivery.

• Circuit Regulation: Capacitors help regulate and control the behavior of electrical circuits and electronic equipment.

• Applications: Used in timing devices (e.g., windshield wipers), filters, and defibrillators.

Definition of Capacitance

Capacitance is a measure of a capacitor's ability to store charge per unit potential difference between its plates.

• Formula: $C = \frac{Q}{\Delta V_C}$

• Units: Farad (F), where $1\,\mathrm{F} = 1\,\mathrm{C}/1\,\mathrm{V}$

For a parallel-plate capacitor:

• $E = \frac{\Delta V}{d}$

• $E = \frac{Q}{\varepsilon_0 A}$

Example: Charging a Capacitor

A 72 pF capacitor is connected to a 10 V battery. Calculate:

• Charge stored: $Q = C \Delta V = 72 \times 10^{-12} \times 10 = 720\,\mathrm{pC}$

• Energy stored: $U = \frac{1}{2} Q \Delta V_C$

Charging Process and Energy Storage

Charging a capacitor involves moving charge from one plate to another, which requires work. The energy stored in the capacitor is equal to the work done by the battery.

• Potential difference for each increment: $\Delta V_i = \frac{q_i}{C}$

• Work for each increment: $\Delta U_i = q_i \Delta V_i$

• Total energy stored: $U = \frac{1}{2} Q \Delta V_C$

Location of Stored Energy

The energy in a charged capacitor is stored in the electric field between its plates. The energy density depends only on the electric field and is independent of the capacitor's geometry. This energy can be transported through space as electromagnetic waves.

Capacitor Behavior in Circuits

Effect of Changing Potential Difference

Doubling the potential difference across a capacitor (by using a battery of double voltage) does not change its capacitance. Capacitance depends only on the physical characteristics of the capacitor (plate area, separation, and dielectric material).

• Key Point: Capacitance remains unchanged when only the voltage is altered.

Effect of Changing Plate Separation

• Isolated Capacitor: If the plates are pulled apart after charging (with the capacitor isolated), the charge remains constant, but the electric field and potential difference change.

• Battery-Connected Capacitor: If the capacitor remains connected to a battery while the plates are separated, the voltage remains constant, but the charge and capacitance change.

Example: Earth-Ionosphere as a Capacitor

The Earth and the ionosphere can be modeled as a spherical capacitor. Given a charge of $5 \times 10^5$ C and a potential difference of $3 \times 10^5$ V:

• Capacitance: $C = \frac{Q}{\Delta V} = \frac{5 \times 10^5}{3 \times 10^5} = 1.67$ F

• Average Electric Field: If the ionosphere is 60 km above the surface, $E = \frac{\Delta V}{d}$

Dielectrics and Capacitance

Introduction to Dielectrics

A dielectric is an insulating material placed between the plates of a capacitor. Its presence increases the capacitance by reducing the effective electric field within the capacitor.

• Dielectric Constant (K): The factor by which the capacitance increases when a dielectric is inserted.

• Formula: $C_{\text{dielectric}} = K C_{\text{vacuum}}$

Dielectric Constants of Common Materials

Material	Dielectric Constant (K)
Vacuum	1 (exactly)
Air	1.00054
Teflon	2.0
Paper	3.0
Pyrex glass	4.8
Cell membrane	9.0
Ethanol	24
Water	80
Strontium titanate	300

Effect of Dielectrics on Energy Storage

• Isolated Capacitor: Inserting a dielectric after disconnecting the capacitor from the circuit reduces the stored energy, since $U \propto E^2$ and the electric field decreases.

• Battery-Connected Capacitor: Inserting a dielectric while the capacitor remains connected to a battery increases the stored energy, as the capacitance increases and the voltage remains fixed.

Key Equations:

• Energy stored: $U = \frac{1}{2} C (\Delta V)^2$

• With dielectric: $C' = K C$

Summary Table: Capacitance and Dielectrics

Scenario	Capacitance	Charge	Voltage	Energy
Isolated, insert dielectric	Increases	Constant	Decreases	Decreases
Battery-connected, insert dielectric	Increases	Increases	Constant	Increases

Key Formulas

• Capacitance: $C = \frac{Q}{\Delta V}$

• Parallel-plate capacitor: $C = \varepsilon_0 \frac{A}{d}$

• Energy stored: $U = \frac{1}{2} Q \Delta V = \frac{1}{2} C (\Delta V)^2$

• With dielectric: $C' = K C$