Electric Potential, Capacitance, and DC Circuits: Structured Study Notes

Electric Potential and Potential Energy

Potential Energy of a Test Charge

The potential energy of a test charge in an electric field is a fundamental concept in electrostatics. When a charge q is moved from point A to point B in a non-uniform electric field \vec{E}(\vec{r}), the work done by the field is equal to the change in the charge's potential energy:

• Work Done: $W_{A \rightarrow B} = q \int_A^B d\vec{r} \cdot \vec{E}(\vec{r})$

• Path Independence: This work is independent of the path taken, ensuring energy conservation.

• Potential Energy: $U(\vec{r}) = q \int_{\vec{r}_0}^{\vec{r}} d\vec{r}' \cdot \vec{E}(\vec{r}') + U(\vec{r}_0)$

Electric Potential

Electric potential is defined as the potential energy per unit charge. It is a scalar quantity and is measured in volts (V):

• Definition: $V(\vec{r}) = \frac{1}{q} U(\vec{r}) = \int_{\vec{r}_0}^{\vec{r}} d\vec{r}' \cdot \vec{E}(\vec{r}') + V(\vec{r}_0)$

• Units: $[V] = \text{V} = \text{J/C} = \text{N} \cdot \text{m/C}$

• Electron-Volt: $1\,\text{eV} = 1.602 \times 10^{-19}\,\text{J}$

Relationship Between Electric Potential and Electric Field

The electric field is related to the spatial variation of the electric potential. The change in potential as a charge moves by $\Delta \vec{r}$ is:

• Potential Change: $\Delta V = \vec{E} \cdot \Delta \vec{r}$

• Component Form: $E_x = \frac{\partial V}{\partial x}$, $E_y = \frac{\partial V}{\partial y}$, $E_z = \frac{\partial V}{\partial z}$

• Gradient Operator: $\vec{\nabla} = \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z}$

• Electric Field: $\vec{E}(\vec{r}) = \vec{\nabla} V(\vec{r})$

Equipotential Surfaces

Properties and Representation

Equipotential surfaces are imaginary surfaces where the electric potential is constant. They are useful for visualizing electric fields:

• Definition: $V(\vec{r}) = \text{const}$ on an equipotential surface

• Field Lines: Electric field lines are always perpendicular to equipotential surfaces

• Density: The denser the equipotential surfaces, the stronger the electric field

• Field Direction: $\vec{E} = \vec{\nabla} V$ cannot be tangential to the surface

Electric Potential of Point Charges and Charge Distributions

Point Charge

The electric potential due to a point charge Q at the origin is:

• Electric Field: $\vec{E}(\vec{r}) = k_e \frac{Q}{r^2} \hat{r}$

• Potential: $V(\vec{r}) = k_e \frac{Q}{r}$

• Equipotential Surfaces: Concentric spheres around the origin

Multiple Charges and Continuous Distributions

• Discrete Charges: $V(\vec{r}) = k_e \sum_j \frac{Q_j}{|\vec{r} - \vec{r}_j|}$

• Continuous Distribution: $V(\vec{r}) = k_e \int \frac{dQ(\vec{r}')}{|\vec{r} - \vec{r}'|}$

Capacitance and Capacitors

Capacitance

Capacitance is the ability of two conducting objects to store charge while remaining electrically neutral. The charge stored is proportional to the potential difference:

• Definition: $Q = CV$

• Units: $[C] = \text{F} = \text{C/V}$

Parallel-Plate Capacitor

• Surface Charge Density: $\sigma = \frac{Q}{A}$

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

• Voltage: $V = \frac{Qd}{\varepsilon_0 A}$

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

Cylindrical Capacitor

• Electric Field: $E(r) = \frac{Q}{2\pi \varepsilon_0 L r}$

• Voltage: $V = \frac{Q}{2\pi \varepsilon_0 L} \ln \frac{b}{a}$

• Capacitance: $C = \frac{2\pi \varepsilon_0 L}{\ln(b/a)}$

Capacitors in Series and Parallel

• Series: $\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$

• Parallel: $C = C_1 + C_2$

Electric Field in Dielectrics and Energy Storage

Energy Density of the Electric Field

The energy stored in a capacitor is the work required to establish the electric field:

• Energy: $U = \frac{CV^2}{2}$

• Energy Density: $u = \frac{U}{Ad} = \frac{\varepsilon_0 E^2}{2}$

Electric Field in Dielectrics

Dielectrics are materials without mobile charge carriers. An external field polarizes the atoms, inducing dipole moments that generate a depolarization field:

• Net Internal Field: $\vec{E}_{\text{int}} = \vec{E}_{\text{ext}} + \vec{E}_{\text{dep}}$

• Dielectric Constant: $K = 1 + \chi$

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

Electric Current, Resistivity, and Resistance

Electric Current and Current Density

Electric current is the rate of charge transfer, and current density is the charge transferred per unit time through a unit surface:

• Current: $I = \frac{dq}{dt}$

• Current Density: $\vec{J} = I/A = en\vec{v}_d$

Mobility, Conductivity, and Resistivity

• Drift Velocity: $\vec{v}_d = \mu \vec{E}$

• Conductivity: $\vec{J} = \sigma \vec{E}$, $\sigma = en\mu$

• Resistivity: $\rho = 1/\sigma$

Resistance and Ohm's Law

• Ohm's Law: $I = V/R$

• Resistance: $R = \rho \frac{L}{A}$

• Units: $[R] = \Omega = \text{V/A}$

Resistivity Temperature Dependence

Resistivity of metals increases with temperature due to increased lattice vibrations:

• Linear Relation: $\rho(T) = \rho(T_0) [1 + \alpha (T - T_0)]$

• Temperature Coefficient: $\alpha$ varies by material

Electromotive Force (emf) and Batteries

Ideal vs. Real Batteries

Real batteries have internal resistance, which affects the voltage delivered to a load:

• Emf: $\mathcal{E} = V_r + V_R = I(r + R)$

• Current: $I = \frac{\mathcal{E}}{r + R}$

• Short Circuit: $R \rightarrow 0$, $I \rightarrow \mathcal{E}/r$

• Open Circuit: $R \rightarrow \infty$, $V_R \rightarrow \mathcal{E}$

Joule Heating (Ohmic Heating)

• Power Dissipated: $P = VI = V^2/R = I^2R$

Resistor Networks and Kirchhoff's Rules

Resistors in Series and Parallel

• Series: $R = R_1 + R_2$

• Parallel: $R = \frac{R_1 R_2}{R_1 + R_2}$

Kirchhoff's Rules

• Junction Rule: $\sum I_i = 0$ (conservation of charge)

• Loop Rule: $\sum \Delta V_i = 0$ (conservation of energy)

• Algorithm: Divide circuit into loops, assign current directions, apply rules, solve equations

RC Circuits

Discharging a Capacitor

• Charge Decay: $Q(t) = \mathcal{E} C e^{-t/\tau}$, $\tau = RC$

• Current Decay: $I(t) = \mathcal{E}/R \cdot e^{-t/\tau}$

Charging a Capacitor

• Charge Growth: $Q(t) = \mathcal{E} C (1 - e^{-t/\tau})$

• Time Constant: $\tau = RC$

Summary Table: Dielectric Constants of Common Materials

Material	Dielectric Constant (K)
Air	1.00006
Benzene	2.28
Oil	2-4
Glass	5-10
Water	80

Additional info: Academic context and expanded explanations were added to ensure completeness and clarity for exam preparation.