PHYS E-1bx: Midterm 2 Study Guide – Circuits, Electromagnetism, and Waves

Circuit Analysis

Resistors in Series and Parallel

Understanding how resistors behave in different configurations is fundamental to analyzing electrical circuits.

• Series Configuration: The total resistance is the sum of individual resistances: $R_{\text{total}} = R_1 + R_2 + \ldots + R_n$

• Parallel Configuration: The reciprocal of the total resistance is the sum of reciprocals: $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}$

• Complex Circuits: Additional wire connections can create combinations of series and parallel elements, requiring stepwise simplification.

Ohm’s Law and Current Flow

• Ohm’s Law: Relates voltage ($V$), current ($I$), and resistance ($R$): $V = IR$

• Current Direction: By convention, current flows from higher to lower electric potential (from positive to negative terminal).

• Magnitude of Current: Determined by the applied voltage and total resistance in the circuit.

Example

• Given a 12 V battery and two resistors (4 Ω and 6 Ω) in series, the total resistance is 10 Ω, so $I = \frac{12\ \text{V}}{10\ \Omega} = 1.2\ \text{A}$.

Transformers and Electromagnetic Induction

Transformer Principles

Transformers use electromagnetic induction to change voltage levels in AC circuits.

• Transformer Equation: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$, where $V$ is voltage and $N$ is the number of turns (subscripts $s$ and $p$ for secondary and primary coils).

• Power Conservation (Ideal): $V_p I_p = V_s I_s$

• Operation: A changing magnetic flux in the primary coil induces an EMF in the secondary coil.

• Applications: Used for voltage step-up (increasing voltage) or step-down (decreasing voltage) in power transmission.

Example

• If $N_s = 200$, $N_p = 100$, and $V_p = 120$ V, then $V_s = 240$ V (step-up transformer).

Motion of Charged Particles in Magnetic Fields

Lorentz Force and Particle Trajectories

Charged particles experience a force when moving through a magnetic field, affecting their motion.

• Lorentz Force: $\vec{F} = q\vec{v} \times \vec{B}$

• Right-Hand Rule: Used to determine the direction of the force on a positive charge; for negative charges, the force is in the opposite direction.

• Circular Motion: The radius of the path is $r = \frac{mv}{|q|B}$, where $m$ is mass, $v$ is speed, $q$ is charge, and $B$ is magnetic field strength.

Example

• An electron ($q = -e$) moving perpendicular to a 0.5 T magnetic field at $2 \times 10^6$ m/s will move in a circle of radius $r = \frac{m v}{e B}$.

Wave Properties and Electromagnetic Waves

Wave Relationships and Graphs

Waves are characterized by their wavelength, frequency, and speed, and can be represented graphically.

• Wave Equation: $v = \lambda f$, where $v$ is speed, $\lambda$ is wavelength, and $f$ is frequency.

• Displacement Graphs: Show how the position of a point on the wave varies with time or position.

• Dipole Antennas: Devices that transmit or receive electromagnetic waves, typically operating at resonant frequencies.

• General Sine Wave Equation: $y(x, t) = A \sin(kx - \omega t + \phi)$, where $A$ is amplitude, $k$ is wave number, $\omega$ is angular frequency, and $\phi$ is phase.

Example

• A radio wave with $f = 100$ MHz and $\lambda = 3$ m travels at $v = 3 \times 10^8$ m/s.

RLC Circuits

Impedance and Resonance

RLC circuits contain resistors (R), inductors (L), and capacitors (C), and exhibit unique behaviors in AC circuits.

• Impedance: $Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$, where $\omega$ is angular frequency.

• Resonance Condition: $\omega_0 = \frac{1}{\sqrt{LC}}$; at resonance, the inductive and capacitive reactances cancel.

• Phase Relationships: The current and voltage can be out of phase, depending on the relative values of $L$ and $C$.

• Energy Oscillations: Energy alternates between the inductor and capacitor during oscillations.

Example

• For $L = 1$ mH and $C = 1$ μF, $\omega_0 = 1/\sqrt{(1 \times 10^{-3})(1 \times 10^{-6})} = 1 \times 10^4$ rad/s.

Standing Waves

Formation and Harmonics

Standing waves result from the superposition of two waves traveling in opposite directions, leading to fixed patterns of nodes and antinodes.

• Nodes: Points of zero amplitude (no motion).

• Antinodes: Points of maximum amplitude.

• Harmonic Frequencies (Strings/Open Pipes): $f_n = \frac{n v}{2L}$, where $n$ is the harmonic number, $v$ is wave speed, and $L$ is length.

Example

• A string of length 1 m with wave speed 100 m/s has fundamental frequency $f_1 = 50$ Hz.

Magnetic Flux and Faraday’s Law

Induction and Lenz’s Law

Changing magnetic flux through a loop induces an electromotive force (EMF), as described by Faraday’s Law.

• Magnetic Flux: $\Phi_B = \vec{B} \cdot \vec{A} = B A \cos \theta$

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

• Lenz’s Law: The direction of induced current opposes the change in magnetic flux.

Example

• If the magnetic field through a loop increases, the induced current creates a field opposing the increase.

Magnetic Forces Generated By Currents

Fields and Forces from Currents

Electric currents generate magnetic fields, which can exert forces on other currents or magnetic materials.

• Magnetic Field from a Long Straight Wire: $B = \frac{\mu_0 I}{2 \pi r}$, where $\mu_0$ is the permeability of free space, $I$ is current, and $r$ is distance from the wire.

• Force on a Wire Segment: $\vec{F} = I \vec{L} \times \vec{B}$, where $\vec{L}$ is the length vector of the wire in the field.

• Net Force on Loops: Analyzing the net force on current loops near other current-carrying wires involves vector addition of forces on each segment.

Example

• A 0.5 m wire carrying 2 A perpendicular to a 0.1 T field experiences a force $F = I L B = 0.1$ N.