Electric Charge, Electric Field, and Capacitance: Study Guide

Electric Charge and Conservation

Conservation of Charge

The principle of conservation of charge states that the total electric charge in an isolated system remains constant over time. This is a fundamental law in physics and applies to all processes involving electric charge.

• Definition: The net charge before and after any physical process is unchanged.

• Example: In chemical reactions, electrons may transfer between atoms, but the total charge remains constant.

Charging Methods

• Charging by Friction: When two materials are rubbed together, electrons may transfer from one to the other, resulting in one object becoming positively charged and the other negatively charged.

• Charging by Conduction: Direct contact between a charged object and a neutral object allows electrons to flow, charging the neutral object.

• Charging by Induction: Bringing a charged object near a neutral conductor causes redistribution of charges within the conductor, which can be separated to create a net charge.

• Example: Rubbing a balloon on hair (friction), touching a charged rod to a metal sphere (conduction), or bringing a charged rod near a metal sphere and grounding it (induction).

Electrostatic Forces

Force Between Charged Particles

The force between two point charges is given by Coulomb's Law:

• Coulomb's Law:

$F = k_e \frac{|q_1 q_2|}{r^2}$

• Where: $k_e$ is the Coulomb constant, $q_1$ and $q_2$ are the charges, $r$ is the distance between them.

• Direction: The force is attractive if charges are opposite, repulsive if they are the same.

Force Between Multiple Charged Particles

• Superposition Principle: The net force on any charge is the vector sum of the forces exerted by all other charges.

• Example: For three charges, calculate the force between each pair and sum the vectors.

Electric Field

Electric Field of a Point Charge

The electric field produced by a point charge is:

$\vec{E} = k_e \frac{q}{r^2} \hat{r}$

• Direction: Away from positive charge, toward negative charge.

Electric Field of Multiple Charges (Superposition)

• Superposition Principle: The total electric field at a point is the vector sum of fields from all charges.

• Example: For two charges, calculate $\vec{E}_1$ and $\vec{E}_2$ at the point, then add.

Electric Field of Distributed Charge

• Continuous Distribution: For a charge distribution, integrate over the distribution:

$\vec{E} = k_e \int \frac{dq}{r^2} \hat{r}$

• Example: Line, surface, or volume charge distributions.

Electric Field in Conductors

• Electrostatic Equilibrium: The electric field inside a conductor is zero; excess charge resides on the surface.

• Example: Hollow metal sphere: no field inside.

Calculating Electric Flux

• Definition: Electric flux through a surface quantifies the number of electric field lines passing through the surface.

$\Phi_E = \int \vec{E} \cdot d\vec{A}$

• Application: Used in Gauss's Law.

Electric Field of Symmetric Distributions

• Spherically Symmetric Distribution: For a sphere, use Gauss's Law:

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$

• Infinite Wire: The field at distance $r$ from an infinite line of charge:

$E = \frac{\lambda}{2\pi \varepsilon_0 r}$

• Infinite Plane: The field near an infinite plane of charge:

$E = \frac{\sigma}{2\varepsilon_0}$

Electric Potential and Energy

Electric Potential Energy (Coulomb Gauge)

• Definition: The energy a charge has due to its position in an electric field.

$U = k_e \frac{q_1 q_2}{r}$

Electric Potential from Point Charges

• Potential at a Point:

$V = k_e \frac{q}{r}$

Electric Potential from Multiple Charges

• Superposition: Sum the potentials from each charge.

$V_{total} = \sum_i k_e \frac{q_i}{r_i}$

Electric Potential from Distributed Charge

• Integration: For continuous distributions:

$V = k_e \int \frac{dq}{r}$

Electric Potential from Conductors

• Conductors: The potential is constant throughout a conductor in electrostatic equilibrium.

Energy Conservation and Potential Changes

Conservation of Energy

• Principle: The total energy (kinetic + potential) in a system is conserved.

• Application: Used to analyze motion of charges in electric fields.

Change in Potential from Electric Field

• Relation: The change in electric potential between two points is:

$\Delta V = - \int_{a}^{b} \vec{E} \cdot d\vec{l}$

Electric Field from Potential

• Gradient: The electric field is the negative gradient of the potential:

$\vec{E} = - \nabla V$

Capacitance and Capacitors

Parallel Plate Capacitors

• Definition: A parallel plate capacitor consists of two plates separated by a distance $d$.

• Capacitance:

$C = \varepsilon_0 \frac{A}{d}$

• Where: $A$ is plate area, $d$ is separation.

Non-Parallel Plate Capacitors

• Varieties: Cylindrical, spherical, and other geometries have different capacitance formulas.

• Example: Cylindrical capacitor:

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

• Where: $L$ is length, $a$ and $b$ are radii.

Capacitors in Networks

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$

• Parallel: $C_{eq} = C_1 + C_2 + ...$

• Application: Used to design circuits with desired capacitance.

Modified Capacitors in Circuits

• Changing Geometry: If plate separation or area changes while connected to a circuit, charge or voltage may change depending on whether the capacitor is isolated or connected.

• Example: If a charged capacitor is disconnected and plate separation increases, voltage increases, charge remains constant.

Dielectric Materials

• Definition: Dielectrics are insulating materials placed between capacitor plates to increase capacitance.

• Effect: Capacitance increases by a factor $\kappa$ (dielectric constant):

$C = \kappa \varepsilon_0 \frac{A}{d}$

• Example: Inserting glass ($\kappa \approx 5$) between plates increases capacitance fivefold.

Capacitors with Dielectric Materials

• Energy Storage: The energy stored in a capacitor is:

$U = \frac{1}{2} C V^2$

• Dielectric Effect: Increases stored energy for a given voltage.

Capacitor Comparison Table

Type	Capacitance Formula	Key Features
Parallel Plate	$C = \varepsilon_0 \frac{A}{d}$	Simple geometry, uniform field
Cylindrical	$C = \frac{2\pi \varepsilon_0 L}{\ln(b/a)}$	Used in coaxial cables
Spherical	$C = 4\pi \varepsilon_0 \frac{ab}{b-a}$	Concentric spheres
With Dielectric	$C = \kappa \varepsilon_0 \frac{A}{d}$	Capacitance increased by $\kappa$

Additional info: Some formulas and examples were inferred for completeness and clarity.