Exam 1 Study Guide: Electricity and Circuits (Chapters 16–19)

Chapter 16: Electric Charge and Electric Field

Coulomb’s Law

Coulomb’s Law describes the force between two point charges. The magnitude of the electrostatic force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them.

• Formula: $F = k_e \frac{|q_1 q_2|}{r^2}$, where $k_e = 8.99 \times 10^9\ \mathrm{N \cdot m^2/C^2}$

• Direction: The force acts along the line joining the charges; like charges repel, unlike charges attract.

• Vector Form: $\vec{F}_{12} = k_e \frac{q_1 q_2}{r^2} \hat{r}$

• Example: Two charges of $+2\ \mu\mathrm{C}$ and $-3\ \mu\mathrm{C}$ separated by $0.5\ \mathrm{m}$ experience a force of $F = k_e \frac{2 \times 10^{-6} \times 3 \times 10^{-6}}{0.25}$.

Electric Field

The electric field at a point is the force per unit positive charge placed at that point.

• Formula: $\vec{E} = \frac{\vec{F}}{q}$

• Due to a Point Charge: $\vec{E} = k_e \frac{q}{r^2} \hat{r}$

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

Electric Field Lines

• Show the direction of the electric field; lines start on positive charges and end on negative charges.

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

Electric Fields and Conductors

• Inside a conductor in electrostatic equilibrium, the electric field is zero.

• Excess charge resides on the surface.

• The field just outside a charged conductor is perpendicular to the surface.

Gauss’s Law

Gauss’s Law relates the electric flux through a closed surface to the charge enclosed by that surface.

• Formula: $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$

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

Chapter 17: Electric Potential and Capacitance

Electric Potential Energy and Potential Difference

Electric potential energy is the energy a charge has due to its position in an electric field. The potential difference (voltage) is the work done per unit charge to move a charge between two points.

• Potential Difference: $\Delta V = V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{s}$

• Unit: Volt (V) = Joule/Coulomb

Relation Between Electric Potential and Electric Field

• The electric field is the negative gradient of the electric potential: $\vec{E} = -\nabla V$

Electric Potential Due to Point Charges

• Formula: $V = k_e \frac{q}{r}$

• For multiple charges, potentials add algebraically: $V_{\text{total}} = \sum_i V_i$

Capacitance

• Capacitance is the ability of a system to store charge per unit potential difference.

• Formula: $C = \frac{Q}{V}$

• Unit: Farad (F)

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

Dielectrics

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

• With Dielectric: $C = \kappa \varepsilon_0 \frac{A}{d}$, where $\kappa$ is the dielectric constant.

Storage of Electric Energy

• Energy stored in a capacitor: $U = \frac{1}{2} C V^2$

Chapter 18: Electric Currents and Resistance

Electric Current

Electric current is the rate of flow of charge through a conductor.

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

• Unit: Ampere (A) = Coulomb/second

Ohm’s Law: Resistance and Resistors

• Ohm’s Law states that the current through a conductor is proportional to the voltage across it: $V = IR$

• Resistance: $R = \frac{V}{I}$

• Unit: Ohm ($\Omega$)

Resistivity

• Resistivity ($\rho$) is a material property that quantifies how strongly a material opposes current flow.

• Formula: $R = \rho \frac{L}{A}$, where $L$ is length and $A$ is cross-sectional area.

Chapter 19: DC Circuits

EMF and Terminal Voltage

Electromotive force (EMF) is the energy provided per charge by a source (like a battery). Terminal voltage is the voltage measured across the terminals of a source when current is flowing.

• EMF ($\mathcal{E}$): The ideal voltage of a source.

• Terminal Voltage: $V_{\text{terminal}} = \mathcal{E} - Ir$, where $r$ is internal resistance.

• Difference: Terminal voltage is less than EMF when current flows due to internal resistance.

Resistors in Series and Parallel

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

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

Kirchhoff’s Rules

• Junction Rule: The sum of currents entering a junction equals the sum leaving it.

• Loop Rule: The sum of potential differences around any closed loop is zero.

Circuits Containing Capacitors in Series and Parallel

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

• Parallel: $C_{\text{eq}} = C_1 + C_2 + \ldots$

RC Circuits (Resistor and Capacitor in Series)

• When a capacitor charges or discharges through a resistor, the voltage and current change exponentially with time.

• Charging: $Q(t) = Q_0 \left(1 - e^{-t/RC}\right)$

• Discharging: $Q(t) = Q_0 e^{-t/RC}$

• Time Constant: $\tau = RC$