Electricity: Charge, Current, Potential Difference, Resistance, DC Circuits and Capacitance
Current, voltage and resistance
Current I = ΔQ/Δt; potential difference V = W/Q; resistance R = V/I. For a metallic conductor at constant temperature, R = ρL/A. In series, R_total = R1 + R2 + … and current is the same. In parallel, 1/R_total = 1/R1 + 1/R2 + … and pd is the same. Thermistors and diodes have non-linear I–V characteristics.
EMF and circuits
EMF ε is the energy transferred per unit charge by the source; terminal pd V = ε − Ir, where r is internal resistance. A potential divider gives V_out = V_in R2/(R1 + R2) for ideal meters. Ammeters go in series with low resistance; voltmeters in parallel with high resistance.
Capacitance
Q = CV; energy stored = ½CV² = ½QV. For charging through a resistor, Q = Q0(1 − e^(−t/RC)); discharging Q = Q0e^(−t/RC). The time constant τ = RC is the time to fall to 1/e of the initial value (about 37%).
Worked example
A 470 μF capacitor charges through 10 kΩ from 12 V. τ = 470 × 10^-6 × 10 × 10^3 = 4.7 s. Time to reach 6.0 V: 6 = 12(1 − e^(−t/τ)), so e^(−t/τ) = 0.5 and t = τ ln2 = 3.3 s. The initial charging current is 12/10^4 = 1.2 mA.
Traps
Using terminal pd as EMF; ignoring internal resistance; adding parallel resistors directly; mixing μF with F; thinking a capacitor blocks current instantly at switch-on; using the discharge equation for charging.