Unit 3 Electric Circuits: Energy, Power, and Capacitors

Power (in circuits)

The rate at which energy is transferred or transformed in a circuit (measured in watts, W).

Power equation (rate form)

P = ΔE/Δt, where ΔE is energy transferred (J) over time Δt (s).

Electric potential energy change for a charge

ΔU = qΔV; moving charge q through potential difference ΔV changes potential energy by qΔV.

Current (definition)

I = Δq/Δt; the rate of flow of charge (coulombs per second, amperes).

Electrical power (general circuit relationship)

P = IV; power associated with an element equals the voltage across it times the current through it.

Passive sign convention (power sign)

If conventional current enters the higher-potential terminal of an element (a voltage drop along current), the element absorbs power (P = +IV); if current enters the lower-potential terminal (a voltage rise), the element delivers power (P is negative under this convention).

Ohm’s law

V = IR for a resistor; voltage across a resistor equals current through it times resistance.

Resistor power formula (current form)

P = I^2R; power dissipated by a resistor when current is known.

Resistor power formula (voltage form)

P = V^2/R; power dissipated by a resistor when voltage across it is known.

Power–current/voltage squaring idea

For a fixed resistor, power scales with the square of current or voltage (e.g., doubling I makes power 4× larger).

Energy from constant power

E = Pt; if power is (approximately) constant over time t, energy converted is P·t.

Series resistors (power comparison rule)

Same current flows through each resistor; Pi = I^2Ri, so the larger resistance dissipates more power in series.

Parallel resistors (power comparison rule)

Same voltage across each branch; Pi = V^2/Ri, so the smaller resistance dissipates more power in parallel.

Power rating

A specification (e.g., 60 W, 1200 W) indicating how much power a device is designed to safely convert under intended operating conditions.

Joule heating

Thermal energy produced when a resistor dissipates electrical power (the usual fate of power in resistors).

Capacitor

A circuit element that stores separated charge and energy in an electric field, typically using two conductors separated by an insulator.

Capacitance

C = Q/V; the amount of charge stored per volt across a capacitor (units: farads, F).

Charge–voltage relationship for a capacitor

Q = CV; stored charge magnitude on a plate equals capacitance times voltage across the capacitor.

Energy stored in a capacitor (voltage form)

U = (1/2)CV^2; energy stored in the electric field of a charged capacitor.

Energy stored in a capacitor (charge form)

U = Q^2/(2C); equivalent capacitor energy expression in terms of Q and C.

Equivalent capacitance (parallel capacitors)

Ceq = C1 + C_2 + …; in parallel, capacitors share the same voltage and total charge adds.

Equivalent capacitance (series capacitors)

1/Ceq = 1/C1 + 1/C_2 + …; in series, capacitors share the same charge and voltages add.

Series capacitors: voltage division rule

In series, each capacitor has the same Q, and Vi = Q/Ci, so the smaller capacitance gets the larger voltage.

DC steady state for an ideal capacitor

After a long time in a DC circuit, an ideal capacitor behaves like an open circuit (current goes to zero).

RC time constant

τ = RC; the characteristic timescale (seconds) for exponential charging/discharging in an RC circuit.