Study Guide: Electric Charge, Electric Field, Electric Potential, Capacitance, and Circuits (Chapters 17–19)

Chapter 17: Electric Charge and Electric Field

Basic Properties of Electric Charge

Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electric or magnetic field. Charges come in two types: positive and negative. Like charges repel each other, while opposite charges attract. The elementary charge, denoted as e, is the magnitude of charge carried by a single proton or electron, with e = 1.60 \times 10^{-19} \text{ C}. Charge is quantized, meaning it only occurs in integer multiples of this elementary charge.

• Conductors: Materials that allow easy movement of charge.

• Insulators: Materials that do not allow easy movement of charge.

Coulomb's Law

Coulomb's law describes the force between two point charges:

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

where $k = \frac{1}{4\pi\epsilon_0} = 8.988 \times 10^9 \text{ N} \cdot \text{m}^2/\text{C}^2$ and $\epsilon_0 = 8.854 \times 10^{-12} \text{ C}^2/\text{N} \cdot \text{m}^2$.

• The force acts along the line joining the two charges.

• Repulsive for like charges; attractive for opposite charges.

Superposition Principle for Forces

When multiple charges exert forces on a single charge, the total force is the vector sum of all individual forces.

• For example, the force on $q_3$ due to $q_1$ and $q_2$ is calculated by summing the vector forces from each charge.

Electric Field

The electric field is a region around a charged object where other charges experience a force. It is defined as:

$\vec{E} = \frac{\vec{F}_0}{q_0}$

For a point charge:

$\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$

The direction of the electric field is away from positive charges and toward negative charges.

Electric Field from Multiple Charges

The total electric field at a point due to several charges is the vector sum of the fields from each charge:

$\vec{E}_{\text{total}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \cdots$

Electric Field Lines

Electric field lines are a visual representation of the electric field. They indicate the direction and strength of the field:

• Lines point away from positive charges and toward negative charges.

• The density of lines indicates the strength of the field.

Electric Flux and Gauss's Law

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

$\Phi_E = EA \cos \phi$

Gauss's law relates the electric flux through a closed surface to the charge enclosed:

$\Phi_E = 4\pi k Q_{\text{encl}}$

Charges Inside Conductors

In electrostatic equilibrium, the electric field inside a conductor is zero, and excess charge resides on the surface. If a cavity is present, the field inside the cavity depends on whether a charge is placed inside.

Chapter 18: Electric Potential and Capacitance

Electric Potential Energy and Work

When a charge moves in an electric field, the field does work on the charge, changing its potential energy:

$W_{a \rightarrow b} = -\Delta U = U_a - U_b$

For a uniform field:

$U = qEy$

For point charges:

$U = k \frac{qq'}{r}$

For multiple charges:

$U = kq' \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \frac{q_3}{r_3} + \cdots \right)$

Electric Potential

Electric potential is the potential energy per unit charge:

$V = \frac{U}{q'}$

For a point charge:

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

For multiple point charges:

$V = k \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \frac{q_3}{r_3} + \cdots \right)$

Equipotential Surfaces

Equipotential surfaces are regions where the electric potential is constant. They are always perpendicular to electric field lines.

Capacitance

Capacitance is the ability of a system to store charge per unit voltage:

$C = \frac{Q}{V_{ab}}$

For a parallel-plate capacitor:

$C = \epsilon_0 \frac{A}{d}$

• Capacitors in series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2}$

• Capacitors in parallel: $C_{eq} = C_1 + C_2$

Energy stored in a capacitor:

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

Dielectric materials increase capacitance by a factor of $k$:

$C = kC_0$

Chapter 19: Current, Resistance, and Directed-Current Circuits

Electric Current

Electric current is the rate of flow of charge:

$I = \frac{\Delta Q}{\Delta t}$

Ohm's Law and Resistance

Ohm's law relates voltage, current, and resistance:

$V = IR$

Resistance depends on material properties:

$R = \rho \frac{L}{A}$

For a source with emf $\epsilon$ and internal resistance $r$:

$I = \frac{\epsilon}{R + r}$

Power in Electric Circuits

Power delivered in a circuit:

$P = IV_{ab} = I^2R = \frac{V_{ab}^2}{R}$

Resistors in Series and Parallel

• Series: $R_{eq} = R_1 + R_2 + R_3 + \cdots$ (same current)

• Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots$ (same voltage)

Kirchhoff’s Rules

Kirchhoff’s rules are used to analyze complex circuits:

• Junction Rule: The sum of currents entering a junction equals the sum leaving ($\sum I = 0$).

• Loop Rule: The sum of voltage changes around a closed loop is zero ($\sum V = 0$).

When calculating, choose a sign convention for current and stick with it (e.g., current into a junction is positive, out is negative).