PHYS 2112 Ch 10 – DC Circuits and Kirchhoff's Rules

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