Practice Final Exam Study Guide: Electricity, Magnetism, and Electromagnetic Waves

Electrostatics and Gauss's Law

Electric Field of a Spherical Shell

Gauss's Law is a fundamental principle used to calculate electric fields produced by symmetric charge distributions. For a nonconducting spherical shell with inner radius R1 and outer radius R2 containing a uniform volume charge density, the electric field at various distances from the center can be determined using Gauss's Law.

• Gauss's Law: The total electric flux through a closed surface is proportional to the enclosed charge: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$

• Regions:

  • For $r < R_1$: No charge is enclosed, so $E = 0$.

  • For $R_1 < r < R_2$: The enclosed charge is proportional to the volume inside radius $r$.

  • For $r > R_2$: The field is as if all charge were concentrated at the center.

• Example: Calculating $E$ at $r = 0.5R_1$, $r = 1.5R_1$, $r = 2R_2$ using the shell's charge density.

Electrostatic Equilibrium and Forces

Suspended Charged Spheres

Charged spheres suspended by threads will repel each other due to electrostatic forces, creating an equilibrium angle. The analysis involves balancing the forces of gravity, tension, and electrostatic repulsion.

• Key Equations:

  • Electrostatic force: $F = \frac{k q^2}{r^2}$

  • Force balance: $\tan \theta = \frac{F_{\text{elec}}}{mg}$

• Example: Two 5.0-g spheres, each with charge $q = 4.0 \times 10^{-8}$ C, suspended from 1.0-m threads, form an angle $\theta$ with the vertical.

Electric Fields and Motion of Charges

Charged Plates and Electron Trajectories

Uniform electric fields between parallel plates can accelerate charged particles. The closest approach of an electron to a plate can be calculated using energy conservation and kinematics.

• Electric Field: $E = \frac{V}{d}$, where $V$ is voltage and $d$ is plate separation.

• Electron Motion: Use $F = eE$ and kinematic equations to find position and velocity.

• Example: An electron with initial velocity $v_0$ enters a field $E = 12,000$ N/C between plates separated by 40 mm.

DC Circuits and Kirchhoff's Laws

Analyzing Complex Circuits

Kirchhoff's laws are used to analyze circuits with multiple loops and branches. The sum of voltages around a closed loop equals zero, and the sum of currents entering a junction equals the sum leaving.

• Kirchhoff's Voltage Law (KVL): $\sum V = 0$ around any closed loop.

• Kirchhoff's Current Law (KCL): $\sum I_{\text{in}} = \sum I_{\text{out}}$ at any junction.

• Example: Calculating currents $I_1$ and $I_2$ in a circuit with resistors and batteries.

Capacitors and RC Circuits

Charging and Discharging Capacitors

Capacitors store energy in electric fields. In RC circuits, the charging and discharging process follows an exponential law.

• Maximum Charge: $Q_{\text{max}} = C V$

• Charging Equation: $Q(t) = Q_{\text{max}} (1 - e^{-t/RC})$

• Discharging Equation: $Q(t) = Q_{\text{max}} e^{-t/RC}$

• Example: A 30-μF capacitor in series with a 25-Ω resistor and a DC battery.

Magnetic Induction and Faraday's Law

Induced EMF in Coils

Changing magnetic flux through a coil induces an electromotive force (EMF) according to Faraday's Law.

• Faraday's Law: $\mathcal{E} = -N \frac{d\Phi_B}{dt}$

• Magnetic Flux: $\Phi_B = B A \cos \theta$

• Example: A coil with $N = 200$, $A = 8.5$ cm², and changing $B$ from 0.020 T to 0.12 T in 0.15 s.

AC Circuits: RLC Series Circuits

Impedance, Power, and Resonance

AC circuits containing resistors (R), inductors (L), and capacitors (C) exhibit impedance and phase relationships. The power factor and average power dissipated are important for understanding energy usage.

• Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$, where $X_L = \omega L$, $X_C = \frac{1}{\omega C}$

• Power Factor: $\cos \phi = \frac{R}{Z}$

• Average Power: $P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos \phi$

• Example: RLC circuit with $R = 15 \Omega$, $L = 25$ mH, $C = 30$ μF, $V = 120$ V, $f = 200$ Hz.

Electromagnetic Waves

Wave Propagation and Field Amplitudes

Electromagnetic waves consist of oscillating electric and magnetic fields that propagate through space. The intensity and field amplitudes are related to the energy carried by the wave.

• Wave Equation: $c = \lambda f$

• Intensity: $I = \frac{1}{2} c \varepsilon_0 E_0^2$

• Field Amplitudes: $E_0$ and $B_0$ are related by $E_0 = c B_0$

• Example: Calculating $E_0$ and $B_0$ for a wave with given intensity.

Radiation Pressure and Acceleration

Electromagnetic waves exert pressure on surfaces, known as radiation pressure. This can cause acceleration of objects such as paper in the path of a laser beam.

• Radiation Pressure: $P = \frac{I}{c}$ for complete absorption

• Acceleration: $a = \frac{F}{m}$, where $F = P \times A$

• Example: A laser beam of power 0.500 mW and diameter 1.0 mm incident on a paper of mass $1.50 \times 10^{-4}$ kg.

HTML Table: Comparison of Circuit Elements

Element	Symbol	Key Equation	Physical Meaning
Resistor	R	$V = IR$	Opposes current, dissipates energy as heat
Capacitor	C	$Q = CV$	Stores energy in electric field
Inductor	L	$V = L \frac{dI}{dt}$	Stores energy in magnetic field

Additional info: These study notes expand upon the practice final exam questions, providing context, definitions, and formulas for key topics in electricity, magnetism, and electromagnetic waves, suitable for college-level physics review.