Physics II: Electric Potential, Capacitance, Current, Resistance, and Circuits – Midterm 2 Study Guide

Electric Potential (Ch 24)

24.1 Electric Potential

Electric potential is a scalar quantity that describes the potential energy per unit charge at a point in an electric field. It is fundamental for understanding energy changes in systems of charges.

• Electric potential energy of two point charges: $U = k \frac{q_1 q_2}{r}$ Where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the charges, and $r$ is the separation distance. Units: Joules (J).

• Electric potential created by a point charge at distance $r$: $V = k \frac{q}{r}$ Units: Volts (V) or J/C.

• Potential energy per unit charge: $U = qV$ For a charge $q$ in potential $V$.

• Work done by electric force: $W = -\Delta U$ If the charge moves: $\Delta U = q \Delta V$

24.2 Equipotential Surfaces

Equipotential surfaces are regions where the electric potential is constant. They are useful for visualizing electric fields and understanding energy changes.

• Definition: A surface where the potential is constant.

• Relationship to electric field: The electric field is always perpendicular to equipotential surfaces.

24.3 Potential Created by a Group of Charged Particles

The principle of superposition allows calculation of the total potential at a point due to multiple charges.

• Superposition principle: $V = V_1 + V_2 + V_3 + \ldots = k \frac{q_1}{r_1} + k \frac{q_2}{r_2} + k \frac{q_3}{r_3} + \ldots$

24.4 Potential Due to an Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. Its potential at a point depends on the dipole moment and the position.

• Formula: $V = \frac{p \cos \theta}{4 \pi \varepsilon_0 r^2}$ Where $p$ is the dipole moment, $\theta$ is the angle, and $r$ is the distance from the dipole center.

• Potential falls off as $1/r^2$ and is proportional to $p$.

24.5 Potential Due to a Continuous Charge Distribution

For continuous distributions, the potential is found by integrating over the charge.

• Formula: $dV = k \frac{dq}{r}$ $V = \int k \frac{dq}{r}$

24.6 Calculating the Field from the Potential

The electric field can be derived from the spatial variation of the electric potential.

• Relationship: $\Delta V = V_f - V_i = - \int \vec{E} \cdot d\vec{s}$ $\vec{E} = - \vec{\nabla} V$

• V decreases along the direction of the field lines.

• Negative charges move toward higher $V$; positive charges move toward lower $V$.

24.7 Electric Potential Energy of a System of Charged Particles

The total potential energy of a system is the sum over all pairs of charges.

• Formula: $U = k \frac{q_1 q_2}{r_{12}} + k \frac{q_1 q_3}{r_{13}} + \ldots + k \frac{q_2 q_n}{r_{2n}} + \ldots$

• $U$ can be positive, negative, or zero depending on the configuration.

24.8 Potential of a Charged Isolated Conductor

Charged conductors have unique properties regarding electric potential and field.

• Inside a conductor: $\vec{E} = 0$, $V$ is uniform.

• Conductors are equipotential regions.

Capacitance (Ch 25)

25.1 Capacitance

Capacitance is a measure of a capacitor's ability to store charge per unit potential difference. It depends only on the geometry of the capacitor.

• Definition: $C = \frac{Q}{V}$ Where $Q$ is the charge and $V$ is the potential difference.

• Capacitance is a property of the capacitor and depends only on its geometry.

25.2 Calculating the Capacitance

Different geometries yield different formulas for capacitance.

• Parallel Plate Capacitor: $C = \varepsilon_0 \frac{A}{d}$

• Spherical Capacitor: $C = 4 \pi \varepsilon_0 \frac{ab}{b-a}$

• Cylindrical Capacitor: $C = 2 \pi \varepsilon_0 L / \ln(b/a)$

25.3 Capacitors in Parallel and in Series

Capacitors can be combined to achieve desired total capacitance.

Configuration	Equivalent Capacitance	Charge	Potential Difference
Parallel	$C_{eq} = \sum C_i$	$Q_{eq} = \sum q_i$	$V_{eq} = V$
Series	$C_{eq} = \frac{1}{\sum \frac{1}{C_i}}$	$Q_{eq} = q$	$V_{eq} = \sum V_i$

• Parallel: Same potential difference, charges may differ.

• Series: Same charge, potential differences may differ.

25.4 Energy Stored in a Capacitor

Capacitors store energy in the electric field between their plates.

• Formula: $U = \frac{CV^2}{2}$

Current and Resistance (Ch 26)

26.1 Electric Current

Electric current is the rate of flow of electric charge through a surface, such as a wire's cross-section.

• Definition: $i = \frac{\Delta q}{\Delta t}$ Units: Amperes (A).

• Current is conserved: $\sum i_{in} = \sum i_{out}$

26.2 Current Density

Current density describes the flow of charge per unit area per unit time.

• Definition: $\vec{J}(\vec{x})$ (Units: A/m2)

• Relation to current: $i = \int_S \vec{J} \cdot d\vec{A}$ If $\vec{J}$ is uniform: $i = J \cdot A$

26.3 Resistance and Resistivity

Resistance quantifies how much a material opposes the flow of electric current. Resistivity is a material property.

• Resistance of a body: $R = \frac{V}{i}$

• Resistivity of a material: $\rho = \frac{E}{J}$

• For a rod of length $L$ and cross-section $A$: $R = \rho \frac{L}{A}$

26.4 Ohm's Law

Ohm's law states a linear relationship between voltage and current for many materials.

• Formula: $V = iR$

• Materials for which $R$ does not change with applied $V$ satisfy Ohm's law.

• Microscopic view: electrons move with a fast random motion plus a slow drift velocity.

26.5 Power

Power in electric circuits is the rate at which energy is transferred or dissipated.

• Formula: $P = iV = \frac{V^2}{R} = i^2 R$

• Energy provided by battery to electrons is dissipated in the resistor.

Circuits (Ch 27)

Electromotive Force (emf) and Power Counting

Electromotive force (emf) is the energy per unit charge supplied by a source such as a battery. Power analysis helps track energy flow in circuits.

• Potential difference between terminals: For ideal emf device: $V = \mathcal{E}$ For real emf device: $V = \mathcal{E} - ir$

• Power provided by emf device: $P_{emf} = i \mathcal{E}$

• Power dissipated in internal resistance: $P_r = i^2 r$

• Power provided to the circuit: $P = P_{emf} - P_r = i \mathcal{E} - i^2 r$

Resistors in Series

Resistors in series share the same current, but the voltage divides among them.

• Equivalent resistance: $R_{eq} = \sum R_i$

• Potential difference in each resistor: $V_i = i R_i$

• Sum of potential differences: $V_{eq} = \sum V_i = i R_{eq}$

Resistors in Parallel

Resistors in parallel share the same voltage, but the current divides among them.

• Equivalent resistance: $R_{eq} = \frac{1}{\sum \frac{1}{R_i}}$

• Current across each resistor: $i_i = \frac{V}{R_i}$

• Sum of currents: $\sum i_i = i_{eq}$ with $i_{eq} = \frac{V}{R_{eq}}$

Solving Circuits

To analyze circuits, apply Kirchhoff's rules and conservation laws.

• Take a "walk along the circuit" to apply the following rules:

• Rules:

  1. Walk through an ideal battery: add $\mathcal{E}$ if you walk from – to +, subtract otherwise.

  2. Walk through a resistor: subtract $iR$ if you walk with the current flow, add $iR$ otherwise.

  3. Loop rule: sum of potential changes along a complete loop is zero.

  4. Apply loop rule to each loop: provides an equation per loop.

  5. Apply conservation of current at junctions.

These equations are sufficient to solve for all currents in the circuit.