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)

$P = \frac{\triangle E}{\triangle t}$, where $\triangle E$ is energy transferred (J) over time $\triangle t$ (s).

Electric potential energy change for a charge

$\triangle U = q\triangle V$; moving charge $q$ through potential difference $\triangle V$ changes potential energy by $q\triangle V$.

Current (definition)

$I = \frac{\triangle q}{\triangle t}$; 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)

$P = \frac{V^2}{R}$; 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 $P\bullet t$.

Series resistors (power comparison rule)

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

Parallel resistors (power comparison rule)

Same voltage across each branch; $P_i = \frac{V^2}{R_i}$, 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

$C = \frac{Q}{V}$; 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)

$U = \frac{1}{2}CV^2$; energy stored in the electric field of a charged capacitor.

Energy stored in a capacitor (charge form)

$U = \frac{Q^2}{2C}$; equivalent capacitor energy expression in terms of $Q$ and $C$.

Equivalent capacitance (parallel capacitors)

$C_{eq} = C_1 + C_2 + \text{…}$; in parallel, capacitors share the same voltage and total charge adds.

Equivalent capacitance (series capacitors)

$1/C_{eq} = 1/C_1 + 1/C_2 + \text{…}$; in series, capacitors share the same charge and voltages add.

Series capacitors: voltage division rule

In series, each capacitor has the same $Q$, and $V_i = \frac{Q}{C_i}$, 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

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