PHYS 2112 Exam – Ch 9-14 (Current, Circuits, Magnetism, Induction, Inductance)

Electric current (definition)

Net amount of charge passing per unit time through an area perpendicular to the flow: I = dq/dt (or Δq/Δt)

SI unit of current

Ampere (A) = 1 coulomb per second (C/s)

Direction of conventional current

The direction positive charge moves (or would move) — opposite to electron flow; from + terminal to − terminal through the external circuit

Resistance (definition)

R = ΔV / I — ratio of voltage across a conductor to the current through it; a measure of opposition to current

Ohm's law

ΔV = IR (V = IR)

SI unit of resistance

Ohm (Ω) = 1 V/A

Ohmic conductor

Resistance stays constant regardless of applied voltage; I vs ΔV graph is a straight line through the origin with slope 1/R

Nonohmic device

Resistance changes with voltage; I vs ΔV graph is nonlinear (e.g. a diode)

Resistance in terms of geometry

R = ρL / A (longer wire → more R; thicker wire → less R)

Resistivity ρ

Intrinsic property of a material at a given temperature; SI unit Ω·m

Conductivity σ

σ = 1/ρ; units (Ω·m)⁻¹ = S/m (Siemens per meter, S = Ω⁻¹)

Resistivity vs temperature

ρ = ρ₀[1 + α(T − T₀)] (also R = R₀[1 + αΔT])

Temperature coefficient of resistivity α

Sets how fast resistivity changes with temperature; units °C⁻¹ or K⁻¹

Wire made 4× longer, same R — new diameter?

2× the original diameter (R ∝ L/d², so d² must go up by 4)

Resistor diameter tripled at same voltage — current?

Current goes up 9× (A ∝ d², so R drops by 9)

Voltage change crossing a resistor in the direction of current

The potential DROPS by IR (current flows from high to low potential)

Electric power (general)

P = IV

Power dissipated in a resistor (3 forms)

P = IV = I²R = V²/R

SI unit of power

Watt (W) = 1 J/s

Superconductivity

Below a critical temperature T_c, a material's resistance drops to exactly zero (Onnes, mercury, ~4.2 K, 1911)

Critical temperature T_c

Temperature below which a material becomes superconducting

Meissner effect

A superconductor expels magnetic fields — lets a magnet levitate above it

Electromotive force (emf, ℰ)

The constant potential difference an ideal battery maintains between its terminals

Battery circuit symbol

Long line = higher-potential (+) terminal; short line = lower-potential (−) terminal

DC circuit

A circuit where current in each branch always flows in the same direction

Resistors in series — what is shared?

Same CURRENT through each resistor

Resistors in series — equivalent resistance

R_eq = R₁ + R₂ + … + R_N (larger than any single resistor)

Resistors in parallel — what is shared?

Same VOLTAGE across each resistor

Resistors in parallel — equivalent resistance

1/R_eq = 1/R₁ + 1/R₂ + … + 1/R_N (smaller than any single resistor)

Two resistors in parallel (shortcut)

R_eq = R₁R₂ / (R₁ + R₂)

Why household lights are wired in parallel

Each gets full voltage, and if one goes out the others keep working

Kirchhoff's junction rule

Sum of currents into a junction = sum of currents out (ΣI = 0)

Junction rule is based on…

Conservation of charge

Kirchhoff's loop rule

Around any closed loop, the algebraic sum of potential changes is zero (ΣΔV = 0)

Loop rule is based on…

Conservation of energy

Loop rule sign: crossing a resistor WITH the current

ΔV = −IR (drop)

Loop rule sign: crossing a resistor AGAINST the current

ΔV = +IR (rise)

Loop rule sign: crossing an emf from − to +

ΔV = +ℰ (rise); from + to − is −ℰ

Kirchhoff problem-solving steps

1) Label loops and nodes 2) Assign current directions 3) Apply junction rule 4) Apply loop rule to closed loops; solve simultaneously

Negative current answer in a Kirchhoff problem means…

The actual current flows opposite to the direction you assumed

RC circuit

Circuit with a resistor and capacitor; capacitor charges or discharges through R

RC time constant

τ = RC (units of seconds)

Charging capacitor — charge vs time

q(t) = Cℰ(1 − e^(−t/RC))

Charging capacitor — current vs time

I(t) = (ℰ/R) e^(−t/RC)

Discharging capacitor — charge vs time

q(t) = Q₀ e^(−t/RC)

Discharging capacitor — current vs time

I(t) = −(Q₀/RC) e^(−t/RC)

After one time constant (charging)

Capacitor reaches ~63% of its final charge

Uses of RC circuits

Timing (wipers, pacemakers, strobe lights, flashbulbs), filters, flashing lights

Magnetic field lines — key property

Always closed loops (no start or end) because there are no magnetic monopoles

Magnetic monopole

An isolated N or S pole — none have ever been found

Magnetic field line direction outside a magnet

Out of the north pole, into the south pole

Field strength from field lines

Strong where lines are close together, weak where far apart; B is tangent to the line

Earth's magnetic field

Earth acts like a bar magnet whose SOUTH pole is near geographic NORTH

Magnetic force on a moving charge

F = qv × B; magnitude F = |q|vB sin θ

Magnetic force on a charge at rest

Zero — only moving charges feel magnetic force

Charge moving parallel to B

Zero magnetic force (sin 0° = 0)

SI unit of magnetic field

Tesla (T) = 1 N/(A·m)

Right-hand rule for F = qv × B

Fingers along v, curl toward B, thumb gives F for a positive charge (reverse for negative)

Symbols ⊙ and ⊗

⊙ = out of the page (arrow tip); ⊗ = into the page (arrow tail)

Work done by magnetic force on a point charge

Zero — force is always perpendicular to velocity, so speed and KE don't change, only direction

Radius of circular motion in a magnetic field

r = mv / (|q|B)

Angular (cyclotron) frequency

ω = |q|B / m (independent of speed)

Period of charged particle in B field

T = 2πm / (|q|B)

Why aurora appear near the poles

Charged particles move nearly parallel to Earth's field there, so the magnetic force (shielding) is ~zero

Applications of circular motion in B fields

Mass spectrometer, cyclotron (e.g. proton therapy)

Force on a current-carrying wire

F = I L × B; magnitude F = ILB sin θ (L points along the current)

Torque on a current loop (N turns)

τ = NIAB sin θ

Magnetic dipole moment of a loop

μ = NIA (direction by right-hand rule); τ = μ × B

Device that runs on torque on a current loop

Electric motor

What creates magnetic fields?

Moving charges (currents). Charges at rest create no magnetic field

Field of a long straight wire

B = μ₀I / (2πr)

Permeability of free space μ₀

4π × 10⁻⁷ T·m/A

Direction of B around a straight wire

Right-hand rule: thumb along current, fingers curl in direction of B (circles around wire)

Field at center of a circular loop (N turns)

B = μ₀NI / (2R)

Right-hand rule for a current loop

Curl fingers with the current; thumb points along B inside the loop

Field inside an ideal solenoid

B = μ₀nI, where n = N/L (turns per length); independent of radius; nearly uniform

Force between two parallel wires

F/l = μ₀I₁I₂ / (2πr)

Parallel wires: same vs opposite current direction

Same direction → attract; opposite directions → repel

Definition of the ampere (parallel wires)

Current in two wires 1 m apart that produces 2 × 10⁻⁷ N per meter of force on each

Biot-Savart law

dB = (μ₀/4π) · I dl × r̂ / r²

Field on the axis of a circular loop

B = μ₀IR² / [2(R² + z²)^(3/2)] (reduces to μ₀I/2R at center)

Ampère's law

∮ B · dl = μ₀ I_enclosed

When is Ampère's law useful?

High-symmetry current distributions (long straight wire, solenoid, toroid)

Motional emf

ℰ = BLv — emf from a conductor of length L moving with speed v perpendicular to B

When is there NO motional emf in a moving rod?

When it moves parallel to B (v × B = 0)

Magnetic flux

Φ_B = BA cos θ (θ between B and the surface normal); generally Φ_B = ∫B·dA

Unit of magnetic flux

Weber (Wb) = 1 T·m²

Faraday's law

ℰ = −N dΦ_B/dt — induced emf equals the rate of change of magnetic flux

Three ways to change flux

Change B, change area A, or change angle θ (rotate the loop)

Lenz's law

The induced current flows so its magnetic field OPPOSES THE CHANGE in flux (not the flux itself)

Lenz's law: flux increasing

Induced field points OPPOSITE to the original field

Lenz's law: flux decreasing

Induced field points in the SAME direction as the original field

Lenz's law steps

1) Find direction of original flux and whether it's increasing/decreasing 2) Find the opposing induced flux 3) Use RHR to get current direction

Electromagnetic induction

Inducing an emf by moving a magnet relative to a coil (or coil relative to a magnet)

Electric generator principle

A coil rotating in a magnetic field has changing flux → induced emf (Faraday's law)

Moving bar on rails with resistor — direction of magnetic force on bar

Opposes the motion (magnetic braking, consistent with Lenz's law)

Mutual inductance M

M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂ — coupling between two nearby circuits

Mutually induced emf

ℰ₂ = −M dI₁/dt ; ℰ₁ = −M dI₂/dt

Self-inductance L

Emf induced in a circuit by its own changing current: ℰ = −L dI/dt ; L = NΦ_B / I

Unit of inductance

Henry (H) = 1 V·s/A