Lec 5

Electric Potential and the Parallel Plate Capacitor

The Electric Field in a Parallel Plate Capacitor

The parallel plate capacitor consists of two large, flat, conducting plates separated by a small distance. When a voltage is applied, one plate accumulates positive charge (+q) and the other negative charge (−q), creating a uniform electric field E between the plates (except near the edges).

• Surface charge density is defined as $\sigma = \frac{q}{A}$, where $q$ is the charge and $A$ is the plate area.

• The electric field between the plates is $E = \frac{\sigma}{\varepsilon_0}$, where $\varepsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \ \mathrm{C^2/(N \cdot m^2)}$).

• The field is uniform as long as you stay away from the edges of the plates.

Equipotential Surfaces and Their Relation to the Electric Field

Equipotential surfaces are regions where the electric potential is constant. In a parallel plate capacitor, these surfaces are parallel to the plates and perpendicular to the electric field lines.

• The relationship between electric field and potential difference is $E = -\frac{\Delta V}{\Delta s}$, where $\Delta V$ is the potential difference and $\Delta s$ is the separation between surfaces.

• For a parallel plate capacitor, $E = \frac{V}{d}$ and $V = Ed$, where $d$ is the distance between the plates.

Capacitors and Dielectrics

Introduction to Dielectrics

A dielectric is an electrically insulating material placed between the plates of a capacitor. Dielectrics increase the capacitor's ability to store charge and allow the plates to be placed closer together without electrical breakdown.

• Dielectrics reduce the risk of breakdown (sparking) compared to air.

• They increase the capacitance by a factor called the dielectric constant ($\kappa$).

The Relation Between Charge and Potential Difference for a Capacitor

The charge stored on a capacitor is directly proportional to the potential difference across its plates:

• $q = CV$, where $C$ is the capacitance in farads (F).

• The larger the capacitance, the more charge can be stored for a given voltage.

• SI unit of capacitance: 1 farad (F) = 1 coulomb/volt.

The Dielectric Constant

The dielectric constant ($\kappa$) is a measure of a material's ability to increase the capacitance of a capacitor compared to a vacuum. When a dielectric is inserted, the capacitance increases by a factor of $\kappa$:

• $C = \kappa C_0$, where $C_0$ is the capacitance without the dielectric.

• The electric field inside the dielectric is reduced: $E = \frac{E_0}{\kappa}$.

• Since $E \leq E_0$, $\kappa \geq 1$ for all materials.

Table: Dielectric Constants of Some Common Substances

Substance	Dielectric Constant, $\kappa$
Vacuum	1
Air	1.00054
Teflon	2.1
Benzene	2.28
Paper (royal gray)	3.3
Ruby mica	5.4
Neoprene rubber	6.7
Methyl alcohol	33.6
Water	80.4

The Capacitance of a Parallel Plate Capacitor

The capacitance of a parallel plate capacitor depends on the area of the plates, the distance between them, and the dielectric material:

• Without dielectric: $C = \varepsilon_0 \frac{A}{d}$

• With dielectric: $C = \kappa \varepsilon_0 \frac{A}{d}$

• Capacitance increases with increasing $\kappa$ and $A$, and decreases with increasing $d$.

Where $A$ is the plate area, $d$ is the separation, $\varepsilon_0$ is the permittivity of free space, and $\kappa$ is the dielectric constant.

Dielectric Breakdown

If the electric field in a capacitor exceeds a certain value, the dielectric material may become conductive, leading to breakdown. The dielectric strength is the maximum electric field a dielectric can withstand without breaking down.

• Example: Mica has a dielectric strength of $1.0 \times 10^8$ V/m, while air has $3.0 \times 10^6$ V/m.

Applications: Capacitance in Computer Keyboards

Capacitive keyboards use the principle of changing capacitance to detect key presses. Each key is associated with a movable plate and a fixed plate, forming a capacitor. Pressing the key changes the separation and thus the capacitance, which is detected electronically.

Effects of Dielectrics: Constant Charge vs. Constant Voltage

Constant Charge (Battery Disconnected)

If a dielectric is inserted after the capacitor is charged and disconnected from the battery, the charge remains constant but the voltage decreases:

• Before: $q = C_0 V_0$

• After: $q = C V$, $C = \kappa C_0$

• Result: $V = V_0 / \kappa$ (voltage decreases for $\kappa > 1$)

Constant Voltage (Battery Connected)

If the capacitor remains connected to the battery while the dielectric is inserted, the voltage stays the same but the charge increases:

• Before: $q_0 = C_0 V$

• After: $q = C V = \kappa C_0 V = \kappa q_0$

• Result: $q$ increases by a factor of $\kappa$

Energy Storage in a Capacitor

Formulas for Energy Stored

The energy stored in a capacitor can be expressed in several equivalent forms:

• $\text{Energy} = \frac{1}{2} qV = \frac{1}{2} CV^2 = \frac{q^2}{2C}$

• For a parallel plate capacitor: $C = \kappa \varepsilon_0 \frac{A}{d}$, $V = Ed$

• Energy density (energy per unit volume): $u = \frac{1}{2} \kappa \varepsilon_0 E^2$

Example: A 10 F capacitor with 3 V across it stores $\frac{1}{2} \times 10 \times 3^2 = 45$ J of energy.

Energy Density in the Electric Field

The energy stored in the electric field can be calculated for any region, such as around a point charge:

• For a point charge: $E = \frac{kq}{r^2}$

• Energy density: $u = \frac{1}{2} \varepsilon_0 E^2$

• Example: At 2.0 cm from a $+40 \mu$C charge, $u = 3.6 \times 10^6$ J/m3

To find the total energy in a shell of radius $r$ and thickness $dr$, multiply the energy density by the shell's volume: $4\pi r^2 dr$.