Capacitors & Dielectrics

Devices and materials for storing electric charge and energy in electric fields.

Definition

A capacitor is a device that stores electric charge and energy in an electric field. It consists of two conductors separated by an insulator (dielectric). Capacitance measures the ability to store charge per unit voltage.

Key point: A capacitor does not create charge — it separates and stores existing charges $Q$ on opposite plates, creating a potential difference $\Delta V$ between them.

Capacitance: $$C = \frac{Q}{\Delta V}$$ where $C$ is in farads (F), $Q$ in coulombs (C), and $\Delta V$ in volts (V).

Symbols for capacitor:

  • Standard symbol: two parallel lines (—||—)
  • Polarized capacitor: one straight line and one curved line (+|()

Parallel-Plate Capacitor

Consists of two conducting plates of area $A$ separated by distance $d$. The space between plates is vacuum or air.

Electric field strength between the plates: $$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{A\varepsilon_0}$$ where $\sigma$ is surface charge density and $\varepsilon_0 = 8.85 \times 10^{-12} \text{ F m}^{-1}$ is the permittivity of free space.

Since $d \ll A$, the electric field $E$ is uniform between the plates and zero elsewhere: $$E = \frac{\Delta V}{d}$$

Capacitance (vacuum/air): $$C = \varepsilon_0 \frac{A}{d}$$

Factors affecting capacitance:

  1. Area of the plate, $A$ — directly proportional ($C \propto A$)
  2. Distance between the plates, $d$ — inversely proportional ($C \propto 1/d$)

Dielectrics

A dielectric is a non-conducting material placed between the plates of a capacitor.

Dielectric constant (relative permittivity): $$\kappa = \frac{\varepsilon}{\varepsilon_0} = \frac{C}{C_0}$$ where $\varepsilon$ is the permittivity of the dielectric material.

Advantages of inserting a dielectric:

  1. Increases capacitance by factor $\kappa$
  2. Increases the maximum operating voltage (higher dielectric strength)
  3. Provides mechanical support between the plates

Dielectric strength is the maximum electric field that can exist in a dielectric without electrical breakdown.

Material Dielectric constant, $\kappa$ Dielectric Strength ($10^6$ V m$^{-1}$)
Paper 3.7 16
Mylar 3.2 7
Rubber 6.7 12
Silicone oil 2.5 15
Nylon 3.4 14
Teflon 2.1 60

Effect on isolated capacitor (battery disconnected): When a dielectric is inserted into a charged capacitor with the battery disconnected, charge $Q_0$ remains constant while the potential difference decreases: $$\Delta V = \frac{\Delta V_0}{\kappa}$$

The capacitance becomes: $$C = \kappa C_0 = \kappa \varepsilon_0 \frac{A}{d}$$

(Subscript 0 denotes parameters for the vacuum-filled capacitor)

Polarization mechanism:

  • Polar molecules are randomly oriented in the absence of an electric field
  • When an external field $\vec{E}_0$ is applied, molecules partially align with the field
  • The polarized dielectric creates an induced electric field $\vec{E}_{ind}$ opposite to $\vec{E}_0$
  • Net field is reduced: $\vec{E} = \frac{\vec{E}_0}{\kappa}$
  • Weaker field means lower voltage for the same charge, resulting in higher capacitance

Energy Storage

A charged capacitor stores electrical potential energy in the electric field between the plates. Energy stored equals the work done to charge the capacitor.

Work is required because electrons must be forced onto a plate that already has electrons (they repel each other). The fuller the plate gets, the more work is required.

From the area under the $\Delta V$ vs $Q$ graph: $$U = W = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}C(\Delta V)^2 = \frac{1}{2}Q\Delta V$$

Energy density in the electric field: $$u = \frac{1}{2}\varepsilon_0 E^2 \quad \text{(vacuum)}$$ $$u = \frac{1}{2}\varepsilon E^2 = \frac{1}{2}\kappa\varepsilon_0 E^2 \quad \text{(with dielectric)}$$

State transitions during charging and discharging:

stateDiagram
    [*] --> Uncharged: Initial state
    Uncharged --> Charging: Connect to battery
    Charging --> FullyCharged: q reaches Qmax
    FullyCharged --> Discharging: Connect to resistor
    Discharging --> Uncharged: q reaches 0
    Discharging --> Charging: Switch to source
    Charging --> Discharging: Switch to load
    FullyCharged --> [*]: Remove circuit
    Uncharged --> [*]: Remove circuit

Key Formulas

Formula Description
$C = \frac{Q}{\Delta V}$ Capacitance definition
$C = \varepsilon_0 \frac{A}{d}$ Parallel plate (vacuum)
$C = \kappa\varepsilon_0 \frac{A}{d}$ With dielectric
$E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{A\varepsilon_0}$ Electric field (vacuum)
$E = \frac{\Delta V}{d}$ Uniform field between plates
$\vec{E} = \frac{\vec{E}_0}{\kappa}$ Net field with dielectric
$\Delta V = \frac{\Delta V_0}{\kappa}$ Potential with dielectric (isolated)
$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ Series combination
$C_{eq} = C_1 + C_2 + \dots$ Parallel combination
$U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}$ Stored energy
$u = \frac{1}{2}\varepsilon_0 E^2$ Energy density (vacuum)
$\tau = RC$ RC time constant
$q(t) = Q_{max}(1 - e^{-t/\tau})$ Charging capacitor
$q(t) = Q_0 e^{-t/\tau}$ Discharging capacitor

Charging and discharging process in an RC circuit:

graph TB
    Start([RC Circuit]) --> Switch{Switch Position}
    Switch -->|Charge| ChargePath[Connect to EMF]
    ChargePath --> Charging[Capacitor charges]
    Charging --> ChargeEnd[VC approaches Vsource]
    Switch -->|Discharge| DischargePath[Connect to resistor]
    DischargePath --> Discharging[Capacitor discharges]
    Discharging --> DischargeEnd[VC decays to zero]
    ChargeEnd -.->|Repeat| Switch
    DischargeEnd -.->|Repeat| Switch

    style Start fill:#e7f5ff,stroke:#1971c2,stroke-width:2px
    style Switch fill:#fff4e6,stroke:#e67700,stroke-width:2px
    style ChargeEnd fill:#d3f9d8,stroke:#2f9e44,stroke-width:2px
    style DischargeEnd fill:#ffe3e3,stroke:#c92a2a,stroke-width:2px

Comparison of series and parallel capacitor combinations:

graph TB
    Title[Capacitor Combinations]

    subgraph series["Series Connection"]
        direction TB
        S_Q[Same charge Q]
        S_Formula["1/Ceq = sum 1/Ci"]
        S_Total[Total C < smallest Ci]
    end

    subgraph parallel["Parallel Connection"]
        direction TB
        P_V[Same voltage V]
        P_Formula["Ceq = sum Ci"]
        P_Total[Total C > largest Ci]
    end

    Title --> series
    Title --> parallel

    style Title fill:#e7f5ff,stroke:#1971c2,stroke-width:3px
    style series fill:#d3f9d8,stroke:#2f9e44,stroke-width:2px
    style parallel fill:#ffe3e3,stroke:#c92a2a,stroke-width:2px

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