Resistance, Ohm’s Law, and Electric Circuits: Study Notes

Electric Energy Storage in Capacitors

Energy Storage Mechanism

Capacitors are devices that store electrical energy by accumulating charge on their plates. The energy is stored in the electric field between the plates, and the battery must do work to move charges onto the plates.

• Potential Energy: When the plates are charged, there is potential energy due to the separation of charges.

• Energy Density: The energy density in the electric field is proportional to the square of the electric field, $E^2$.

• Parallel Plate Capacitor: For a parallel plate capacitor, $E = \frac{\sigma}{\epsilon_0}$, where $\sigma$ is the surface charge density and $\epsilon_0$ is the permittivity of free space.

• Electrostatic Energy Stored:

  • $U = \frac{\epsilon_0}{2} \left( \frac{\sigma}{\epsilon_0} \right)^2 A d$

  • $U = \frac{1}{2} \frac{Q^2}{C}$

  • In terms of potential difference: $U = \frac{1}{2} C V^2$

Example

If a capacitor with capacitance $C$ is charged to a voltage $V$, the energy stored is $U = \frac{1}{2} C V^2$.

Dielectrics

Properties and Effects

A dielectric is an insulating material (e.g., glass) that can be polarized when placed in an electric field. The positive and negative charges in the dielectric become slightly separated, inducing surface charges.

• Polarization: The degree of polarization depends on the material.

• Induced Charge: When a dielectric is inserted between capacitor plates, induced charges create an electric field opposite to the original field, partially canceling it.

• Capacitance Increase: If the capacitor is held at constant potential, more charge must be added to compensate for the reduced field, increasing the capacitance.

• Capacitance with Dielectric: $C = K C_0$, where $K$ is the dielectric constant and $C_0$ is the original capacitance.

• Energy Storage: The battery does more work to add charge, so the total energy stored increases.

Example

Inserting a dielectric with $K = 2$ doubles the capacitance of a parallel plate capacitor.

Capacitors in Series and Parallel

Configurations

Capacitors can be connected in series or parallel, affecting the total capacitance.

• Parallel: All top plates are connected to one terminal, all bottom plates to the other. The total capacitance is the sum of individual capacitances.

• Series: Capacitors are stacked, with each connected to the next. The reciprocal of the total capacitance is the sum of reciprocals of individual capacitances.

Configuration	Total Capacitance Formula
Parallel	$C_{eq} = C_1 + C_2 + \ldots$
Series	$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$

Electric Current and Ohm’s Law

Definition of Electric Current

Electric current is the net flow of charge through a cross-sectional area per unit time. The SI unit is the ampere (A), equivalent to one coulomb per second.

• Direction: By convention, current flows from high potential to low potential, even though electrons move in the opposite direction in conductors.

• Carriers: In conductors, electrons carry current; in plasmas and solutions, ions may also carry current.

• Formula: $I = \frac{\Delta Q}{\Delta t}$

Ohm’s Law

Ohm’s Law relates the potential difference across a conductor to the current flowing through it and its resistance.

• Formula: $V = I R$

• Resistance ($R$): The constant of proportionality, measured in ohms ($\Omega$).

• Empirical Law: Larger potential difference results in larger current.

Example

If a resistor of $10\,\Omega$ is connected to a $5\,\text{V}$ battery, the current is $I = \frac{V}{R} = 0.5\,\text{A}$.

Resistance and Resistivity

Factors Affecting Resistance

Resistance depends on both the geometry and the material of the conductor.

• Geometry:

  • Cross-sectional Area ($A$): Larger area means lower resistance.

  • Length ($L$): Longer wire means higher resistance.

  • Formula: $R \sim \frac{L}{A}$

• Material:

  • Resistivity ($\rho$): A property of the material, found in tables.

  • Formula: $R = \rho \frac{L}{A}$

Example

Compare a copper wire and a carbon cylinder: Copper has much lower resistivity, so its resistance is much lower for the same geometry.

Electric Circuits

Basic Concepts

An electric circuit is a collection of components (resistors, capacitors, voltage sources) connected to allow current to flow.

• Battery: Supplies electrical energy and maintains steady current.

• Current Flow: Current flows from positive to negative outside the battery, and from negative to positive inside the battery.

• Resistors: Used to control the amount of current flowing through devices.

Example

Adding a resistor to a circuit with a light bulb allows control of brightness by limiting current.

Resistors in Circuits: Series vs. Parallel

Series Configuration

Resistors in series have the same current flowing through each. The total voltage drop is the sum of individual drops.

• Formula: $R_{eq} = R_1 + R_2 + \ldots$

Parallel Configuration

Resistors in parallel have the same voltage across each, but the current splits among the paths.

• Formula: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

Configuration	Total Resistance Formula
Series	$R_{eq} = R_1 + R_2 + \ldots$
Parallel	$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

Energy in Electric Circuits

Energy Dissipation

As charges move from high to low potential, potential energy is converted to kinetic energy. In resistive wires, collisions cause energy to dissipate as heat.

• Power: The rate at which electrical energy is converted to heat is called power.

• Formula: $P = I V$

• Units: 1 Watt ($\text{W}$) = 1 Joule per second ($\text{J/s}$)

Example

A circuit with $I = 2\,\text{A}$ and $V = 5\,\text{V}$ dissipates $P = 10\,\text{W}$ of power as heat.