Core Topics in Electromagnetism and Circuits

Polarization of Light (Malus’s Law)

Polarization describes the orientation of the electric field vector in a light wave. When unpolarized light passes through a polarizer, only the component aligned with the polarizer's axis is transmitted.

• Malus’s Law: The intensity of polarized light after passing through a polarizer at angle θ to the initial polarization is given by:

• Unpolarized Light: The first polarizer transmits half the intensity:

• Multiple Polarizers: For each subsequent polarizer, use the angle between adjacent axes.

• Crossed Polarizers: Two polarizers at 90° block all light, but inserting a third at an intermediate angle allows some transmission.

• Sanity Check: Final intensity should be less than for unpolarized input.

Example: Three polarizers at 0°, 40°, and 90°: angles between are 40° and 50°, not 90°.

DC Circuits and Resistor Networks

DC circuits involve resistors, batteries, and the analysis of current and voltage using Ohm’s Law and Kirchhoff’s Rules.

• Series:

• Parallel:

• Total Current:

• Kirchhoff’s Laws:

  • Junction Rule:

  • Loop Rule: around any closed loop

• Internal Resistance: For a real battery,

• Power:

Example: Two resistors in parallel only if they share both nodes.

Magnetic Field of a Wire

A current-carrying wire produces a magnetic field whose strength and direction can be determined using Ampère’s Law and the right-hand rule.

• Long Straight Wire: , where is the perpendicular distance from the wire.

• Direction: Right-hand rule: thumb in direction of current, fingers curl in direction of .

• Constant: T·m/A

Example: Find at 5 cm from a wire carrying 2 A.

Faraday’s Law and Lenz’s Law

Changing magnetic flux through a loop induces an electromotive force (EMF) and current, with the direction given by Lenz’s Law.

• Faraday’s Law: ,

• Lenz’s Law: The induced current creates a magnetic field opposing the change in flux.

• Time-Dependent Fields: If is given,

• Flux Change by Area or Angle: can change if , , or changes.

Example: A loop pulled out of a magnetic field region experiences a decreasing flux, inducing a current that tries to maintain the original flux.

RLC and AC Circuits

Alternating current (AC) circuits with resistors (R), inductors (L), and capacitors (C) exhibit frequency-dependent behavior characterized by impedance and phase relationships.

• Angular Frequency:

• Inductive Reactance:

• Capacitive Reactance:

• Impedance:

• Peak Current:

• Phase Angle:

• Resonance: ,

Example: At resonance, , , and current is maximized.

Cyclotron Motion

Charged particles moving perpendicular to a uniform magnetic field undergo circular motion due to the Lorentz force.

• Radius:

• Velocity from Kinetic Energy:

• Cyclotron Frequency:

Example: An electron with 1 keV energy in a 0.1 T field: convert eV to J, find , then .

Magnetic Torque on a Current Loop

A current-carrying loop in a magnetic field experiences a torque that tends to align its magnetic moment with the field.

• Magnetic Moment:

• Torque:

• Equilibrium: (if balancing a mass)

Example: A coil with area and turns in field, carrying , experiences maximum torque when .

EM Wave Intensity and Power

Electromagnetic (EM) waves carry energy, which can be quantified by intensity and power. The Poynting vector describes the energy flow per unit area.

• Intensity: ,

• Radiation Pressure: (absorbed), (reflected)

• Energy Density:

• Relationship:

Example: Solar intensity at 1 AU: W/m², m.

Expanded Topics in Electromagnetism and Circuits

RC and LR Circuit Transients

Capacitors and inductors respond to changes in voltage and current over characteristic time scales.

• RC Circuit Time Constant:

• Discharging Capacitor: ,

• Charging Capacitor: ,

• LR Circuit Time Constant:

• Current Growth:

• Current Decay:

Comparison Table:

LC Oscillations (Undamped)

An LC circuit oscillates energy between the electric field of the capacitor and the magnetic field of the inductor.

• Angular Frequency:

• Charge Oscillation:

• Energy Conservation:

• Total Energy: (constant)

Self-Inductance and Inductor EMF

Inductors resist changes in current, producing an EMF proportional to the rate of change of current.

• Inductor EMF:

• Solenoid Inductance:

• Energy Stored:

• Energy Density:

Ampère’s Law Applications

Ampère’s Law relates the integrated magnetic field around a closed loop to the current passing through the loop.

• Solenoid (inside):

• Inside Thick Conductor (r < R):

• Outside Cylinder (r > R):

Motional EMF and Sliding Rail

Moving a conductor through a magnetic field induces an EMF and current, with associated mechanical and electrical power.

• Motional EMF:

• Current:

• Retarding Force:

• Mechanical Power:

Induced Electric Field

A changing magnetic field induces a non-conservative electric field, as described by Faraday’s Law in integral form.

• Faraday’s Law (integral):

• Inside Solenoid (r < R):

• Outside Solenoid (r > R):

Transformers

Transformers use mutual induction to transfer energy between circuits, changing voltage and current according to the turns ratio.

• Voltage Ratio:

• Power Conservation (ideal):

• Step-up: (voltage increases, current decreases)

• Step-down: (voltage decreases, current increases)

Magnetic Force on Charged Particle (Lorentz Force)

A charged particle in electric and magnetic fields experiences a force given by the Lorentz force law.

• Lorentz Force:

• Velocity Selector:

• Direction: Use right-hand rule; for negative charges, reverse the direction.

Force Between Parallel Wires

Parallel current-carrying wires exert forces on each other due to their magnetic fields.

• Force per Unit Length:

• Direction: Same direction currents attract; opposite directions repel.

Force on a Current-Carrying Wire in External B

A wire carrying current in a magnetic field experiences a force perpendicular to both the current and the field.

• Force: ,

• Levitation: (for magnetic levitation)

B-field at Center of Circular Loop/Coil

The magnetic field at the center of a current loop is directed along the axis perpendicular to the plane of the loop.

• Single Loop:

• N-turn Coil:

EM Wave Properties and Relationships

Electromagnetic waves are characterized by their speed, wavelength, frequency, and the relationship between electric and magnetic fields.

• Speed:

• Wavenumber:

• Angular Frequency:

• Field Relationship:

• Wave Equation: ,

• Energy Densities: , ,

RC Filters (High-Pass / Low-Pass)

RC circuits can act as frequency filters, passing or blocking signals depending on frequency.

• Cutoff Angular Frequency:

• Cutoff Frequency:

• Low-pass: Output across capacitor; passes low frequencies.

• High-pass: Output across resistor; passes high frequencies.

Special Topics

Displacement Current (Ampère-Maxwell Law)

Maxwell added the displacement current term to Ampère’s Law to account for changing electric fields, ensuring consistency with charge conservation.

• Ampère-Maxwell Law:

• Displacement Current:

• Physical Meaning: A changing electric field (e.g., between capacitor plates) produces a magnetic field like a real current.

Galilean Field Transformations

At low velocities, electric and magnetic fields transform between reference frames according to Galilean approximations.

• Transformations:

• Physical Meaning: A pure magnetic field in one frame appears as a combination of electric and magnetic fields in another.

Power Dissipation and Energy Storage

Electrical circuits dissipate and store energy in resistors, capacitors, and inductors.

• Resistor Power:

• Battery Power:

• Capacitor Energy:

• Inductor Energy:

• Field Energy Densities: ,

• Efficiency: (for internal resistance )

Quick Reference Table: Key Equations

Constants: T·m/A, C²/N·m², m/s, C

Additional info: This guide covers all major equations and concepts for electromagnetism and circuits relevant to a second-semester introductory physics course, including DC/AC circuits, magnetic fields, induction, EM waves, and energy relationships.