Electrical Potential, Electrical Energy, and Capacitance – Study Notes

Chapter 17: Electrical Potential, Electrical Energy, and Capacitance

Electrical Potential Energy

Electrical potential energy is the energy stored due to the position of a charge in an electric field. It is analogous to gravitational potential energy, but involves electric forces instead of gravitational forces.

• Definition: The change in electric potential energy between two points A and B is given by: $PE_a - PE_b = -W_{ba}$ where $W_{ba}$ is the work done by the electric force as the charge moves from point b to point a.

• Comparison to Gravitational Potential Energy: Gravitational: $W = F d = mgh = \Delta U$ Electrical: $W = F d = qEd = \Delta U_E$

• Work in Electric Field: $W_A = F d = \Delta U_E$ $W_A = (qE)d$ $W_E = -(qE)d$ $W_A = \Delta U_E = qEd$

• Example: Moving a charge between two parallel plates (as shown in the diagram) requires work, which is stored as electrical potential energy.

Voltage (Electric Potential Difference)

Voltage is the work done per unit charge to move a charge between two points in an electric field. It is also called electrical potential difference.

• Definition: $V = \frac{W}{q} = -\frac{qE \cdot d}{q} = -Ed$

• Units: The unit of voltage is the volt (V), where $1\,\text{V} = 1\,\text{J}/\text{C}$.

• Reference Point: Ground is typically considered to be at $0\,\text{V}$, and the negative terminal of a battery is often connected to ground.

• Example: Batteries are labeled with their voltage relative to ground (e.g., $0.0\,\text{V}$, $+0.75\,\text{V}$, $+1.5\,\text{V}$).

Relationship Between Electrical and Gravitational Potential Energy

Both electrical and gravitational potential energies are forms of potential energy associated with a force field. The equations for work and energy are structurally similar:

• Gravitational: $W = F_g d = mgh = \Delta U$

• Electrical: $W = F_e d = qEd = \Delta U_E$

• Key Point: In both cases, work is required to move an object against the field, and this work is stored as potential energy.

Equipotential Lines and Surfaces

Equipotential lines (or surfaces in 3D) are locations where the electric potential is constant. They are always perpendicular to electric field lines.

• Definition: An equipotential surface is a surface on which every point has the same electric potential.

• Properties:

  • Equipotential lines are always perpendicular to electric field lines.

  • The closer the equipotential lines, the stronger the electric field.

• Example: In a parallel plate capacitor, the plates themselves are equipotential surfaces.

Electric Potential Due to Point Charges

The electric potential at a point due to a single point charge is derived from Coulomb's Law.

• Formula: $V = k \frac{Q}{r}$ where $k = 9.0 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$, $Q$ is the charge, and $r$ is the distance from the charge.

• Superposition Principle: The total potential from multiple point charges is the algebraic sum of the potentials from each charge: $V_{total} = V_1 + V_2 + V_3 + \ldots$

• Note: Unlike electric field vectors, potentials are scalars and can be added directly, but the sign of each charge must be considered.

• Example: Calculating the potential at a point due to several charges at different distances.

Electric Dipoles and Dipole Potential

An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment is a vector pointing from the negative to the positive charge.

• Dipole Moment: $\vec{p} = Q \vec{L}$

• Potential Due to a Dipole: For points far from the dipole ($r \gg L$): $V = k \frac{p \cos \theta}{r^2}$

• Units: Dipole moment is measured in coulomb-meters (Cm) or debye (D), where $1\,\text{D} = 3.33 \times 10^{-30}\,\text{Cm}$.

• Example: Water molecule has a dipole moment of $1.8\,\text{D}$.

Capacitance

Capacitance is the ability of a system to store electric charge. It is defined as the ratio of the charge stored to the potential difference across the system.

• Definition: $C = \frac{Q}{V}$

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

• Typical Values: Most capacitors have values in the microfarad ($\mu$F) or picofarad (pF) range, as 1 F is very large.

• Example: A parallel plate capacitor is a common type, consisting of two plates separated by a distance $d$.

Parallel Plate Capacitor

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

• Formula: $C = \varepsilon_0 \frac{A}{d}$ where $\varepsilon_0 = 8.85 \times 10^{-12}\,\text{C}^2/\text{N}\cdot\text{m}^2$ is the permittivity of free space, $A$ is the area, and $d$ is the separation.

• Dielectrics: Inserting a dielectric increases capacitance: $C = K \varepsilon_0 \frac{A}{d}$ where $K$ is the dielectric constant of the material.

• Example: Paper ($K \approx 3.7$), Water ($K \approx 80$), Glass ($K \approx 4.7$).

Energy Stored in a Capacitor

Capacitors store energy in the electric field between their plates. The energy stored depends on the charge and voltage.

• Formula:

  • $U = \frac{1}{2} Q V$

  • $U = \frac{1}{2} C V^2$

  • $U = \frac{1}{2} \frac{Q^2}{C}$

• Energy Density: For a parallel plate capacitor, the energy density (energy per unit volume) is: $u = \frac{1}{2} \varepsilon_0 E^2$

• Example: A camera flash uses a capacitor to store energy for a rapid discharge.

Dielectrics and Their Effect on Capacitance

Dielectrics are insulating materials placed between the plates of a capacitor. They increase the capacitance by reducing the effective electric field.

• Dielectric Constant ($K$): Ratio of the permittivity of the material to that of free space: $K = \varepsilon / \varepsilon_0$

• Effect: Inserting a dielectric increases capacitance by a factor of $K$.

• Microscopic Explanation: Dielectrics become polarized in an electric field, inducing dipoles that partially cancel the field inside the material.

• Example Table:

Material	Dielectric Constant (K)
Vacuum	1.00000
Air	1.0006
Paper	3.7
Glass	4.7
Water	80
Hard Rubber	2.8
Ceramics	~10,000

Applications of Capacitors

Capacitors are widely used in electronic circuits for various purposes.

• Storing charge and energy (short-term)

• Converting AC to DC

• Surge protection

• Computer memory and keyboards

• High current sources (e.g., camera flashes)

Summary Table: Key Equations

Concept	Equation
Electric Potential (Point Charge)	$V = k \frac{Q}{r}$
Capacitance (Parallel Plate)	$C = \varepsilon_0 \frac{A}{d}$
Capacitance (with Dielectric)	$C = K \varepsilon_0 \frac{A}{d}$
Energy Stored in Capacitor	$U = \frac{1}{2} C V^2$
Energy Density	$u = \frac{1}{2} \varepsilon_0 E^2$
Dipole Potential (far field)	$V = k \frac{p \cos \theta}{r^2}$

Additional info:

• Some equations and explanations have been expanded for clarity and completeness.

• Examples and applications have been added to illustrate key concepts.

• Tables have been reconstructed and summarized for study purposes.