Physics with Calculus: Study Notes on Chapters 21–24 (Electricity and Capacitance)

Chapter 21: Electric Charge and Electric Field

Insulators and Conductors

Materials can be classified based on their ability to conduct electric charge:

• Conductors: Materials (e.g., metals) in which electric charges move freely.

• Insulators: Materials (e.g., glass, rubber) in which electric charges do not move freely.

Coulomb’s Law

Coulomb’s Law describes the force between two point charges:

• The force is proportional to the product of the charges and inversely proportional to the square of the distance between them.

Equation:

$F = k \frac{Q_1 Q_2}{r^2} \hat{r} = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2} \hat{r}$

• k is Coulomb’s constant: $k = 8.99 \times 10^9 \ \mathrm{N \cdot m^2 / C^2}$

• $\varepsilon_0$ is the permittivity of free space.

Electric Field

The electric field at a point is the force per unit charge at that point:

$\vec{E} = \frac{\vec{F}}{q}$

For a point charge:

$\vec{E} = k \frac{Q}{r^2} \hat{r} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{r}$

For a continuous charge distribution:

$\vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{dQ}{r^2} \hat{r}$

Superposition Principle

• The net electric field is the vector sum of the fields produced by each charge.

Vectors in Electric Field Calculations

• Electric fields and forces are vector quantities; direction must be considered in calculations.

Dipole Moment

• A dipole consists of two equal and opposite charges separated by a distance l.

• The dipole moment is a vector from negative to positive charge:

$\vec{p} = Q \vec{l}$

• Do not confuse p (dipole moment) with p for momentum.

Torque and Potential Energy of a Dipole

• Torque on a dipole in an electric field:

$\vec{\tau} = \vec{p} \times \vec{E}$

• Potential energy of a dipole in an electric field:

$U = -\vec{p} \cdot \vec{E}$

Equations of Motion with Electric Fields

• Combine electric field and force equations with kinematic equations to solve motion problems.

Newton’s Second Law:

$\vec{F} = m \vec{a} = q \vec{E}$

Kinematic equations (for constant acceleration):

$v = v_0 + at$

$x = x_0 + v_0 t + \frac{1}{2} a t^2$

$v^2 = v_0^2 + 2a x$

Chapter 22: Gauss’s Law

Electric Flux

Electric flux through a surface measures the number of electric field lines passing through that surface:

$\Phi_E = \vec{E} \cdot \vec{A} = EA \cos \theta$

For a non-uniform field or curved surface:

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

Gauss’s Law

Gauss’s Law relates the electric flux through a closed surface to the charge enclosed:

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

• Applies to closed surfaces (surfaces that enclose a volume).

• Useful for calculating electric fields with high symmetry (spherical, cylindrical, planar).

Chapter 23: Electric Potential

Potential and Potential Energy

• Electric potential difference (voltage) is the work done per unit charge to move a charge between two points.

$\Delta V = V_b - V_a = - \int_a^b \vec{E} \cdot d\vec{l}$

Relationship to potential energy:

$\Delta U = q \Delta V$

Potential Due to Point Charges and Continuous Distributions

For a point charge:

$V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r}$

For a continuous charge distribution:

$V = \frac{1}{4\pi\varepsilon_0} \int \frac{dQ}{r}$

Electric Field from Potential

The electric field is the negative gradient of the potential:

$\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right)$

Electron Volt (eV)

• The electron volt is a unit of energy: $1\ \mathrm{eV} = 1.6 \times 10^{-19}\ \mathrm{J}$

Chapter 24: Capacitance, Dielectrics, and Energy Storage

Capacitance

• Capacitance (C) is the ability of a system to store charge per unit voltage.

$C = \frac{Q}{V}$

• Unit: Farad (F)

Capacitors in Series and Parallel

Configuration	Charge (Q)	Voltage (V)	Equivalent Capacitance (CT)
Series	Same	Add	$\frac{1}{C_T} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$
Parallel	Add	Same	$C_T = C_1 + C_2 + \ldots$

Energy Stored in a Capacitor

The energy stored in a capacitor is equal to the work done to charge it:

$U = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C}$

For a parallel-plate capacitor:

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

Energy stored in the electric field (energy density):

$u = \frac{U}{\text{Volume}} = \frac{1}{2} \varepsilon_0 E^2$

Changes in Capacitors

• If a capacitor remains connected to a battery, V is constant and Q will change if C changes.

• If disconnected from a battery, Q is constant and V will change if C changes.

Dielectrics

• A dielectric is an insulating material placed between the plates of a capacitor to increase its capacitance.

• The dielectric constant (K) is always greater than 1 and has no units.

Effects of a dielectric:

• Capacitance increases: $C = K C_0$

• Permittivity increases: $\varepsilon = K \varepsilon_0$

• Electric field decreases: $E = \frac{E_0}{K}$

Summary Table: Effects of Dielectrics

Property	Without Dielectric	With Dielectric (K)
Capacitance	$C_0$	$K C_0$
Permittivity	$\varepsilon_0$	$K \varepsilon_0$
Electric Field	$E_0$	$\frac{E_0}{K}$

Example: Parallel-Plate Capacitor with Dielectric

• Suppose a parallel-plate capacitor with area A and separation d is filled with a dielectric of constant K:

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

The energy stored becomes:

$U = \frac{1}{2} K \varepsilon_0 \frac{A}{d} V^2$

Additional info: The above notes expand on the brief points in the original file, providing definitions, equations, and context for each topic. Tables have been reconstructed for clarity. All equations are provided in LaTeX format as required.