Circuits Part 2: EMF & Capacitors

EMF and Real Batteries

A real battery is an ideal EMF in series with an internal resistance: $V_{terminal} = \varepsilon - Ir$, so terminal voltage sags as more current is drawn. Ammeters go in series with almost no resistance; voltmeters go in parallel with very high resistance.

Capacitors

$C = \frac{Q}{V} \qquad C = \kappa\varepsilon_0\frac{A}{d} \qquad U = \tfrac{1}{2}CV^2$ A dielectric raises capacitance. With the battery connected, voltage is fixed; disconnected, charge is fixed. Capacitors combine opposite to resistors: directly in parallel, reciprocally in series.

RC Circuits

An uncharged capacitor acts like a wire the instant a switch closes and like a break once charged. The time constant $\tau = RC$ sets the pace: about 63% charged after one $\tau$, essentially full after about $5\tau$.

Bioelectricity

The cell membrane is a capacitor (the bilayer) in parallel with resistors (the ion channels). Myelin lowers membrane capacitance and raises membrane resistance, speeding conduction. A defibrillator stores energy in a capacitor and releases it in a brief pulse.

Key Takeaways

• Terminal voltage is below the EMF whenever current flows out of a battery.
• Decide first whether Q or V is held constant, then choose the energy formula.
• Capacitor combination rules are the reverse of resistor rules.
• Steady state: no current through a capacitor branch.