Transient State: Current and voltage are changing rapidly, inductor's properties are time-dependent. | Steady State: Current and voltage are stable, inductor acts like a wire with no resistance.

Immediately After: Inductor resists current change, current starts at zero (if initially zero). | After a Long Time: Inductor acts like a short circuit, current reaches its maximum value ($I = \frac{\mathcal{E}}{R}$).

Resistors: Dissipate energy as heat, voltage is proportional to current (Ohm's Law). | Inductors: Store energy in a magnetic field, voltage is proportional to the rate of change of current.

Charging: Current increases exponentially from zero towards a maximum value. | Discharging: Current decreases exponentially from an initial value towards zero.

Increasing Resistance: Decreases the time constant ($\tau = \frac{L}{R}$), leading to a faster response. | Increasing Inductance: Increases the time constant, leading to a slower response.

1. Identify a closed loop in the circuit. 2. Sum the voltage drops across each element in the loop, considering the sign. 3. Set the sum equal to zero: $\mathcal{E} - IR - L \frac{dI}{dt} = 0$.

1. Identify the inductance (L) and the equivalent resistance ($R_{eq}$) in the circuit. 2. Use the formula: $\tau = \frac{L}{R_{eq}}$.

1. Apply Kirchhoff's loop rule to obtain a differential equation. 2. Solve the differential equation for the current as a function of time, I(t). 3. Analyze how the current, voltage, and energy change over time using exponential functions.

1. After a long time, treat the inductor as a short circuit (a wire with zero resistance). 2. Calculate the current through the circuit using Ohm's law, considering only the battery's emf and the total resistance.

1. Recognize that the current through the inductor cannot change instantaneously. 2. If the initial current is zero, the current remains zero immediately after closing the switch.

A property of a circuit element that opposes changes in current, measured in henries (H).

A circuit element that opposes the flow of current, dissipating energy as heat.

The time it takes for the current in an LR circuit to reach approximately 63% of its final value during charging or to decrease to 37% of its initial value during discharging.

The initial period after a change (like closing a switch) when current and voltage are changing rapidly.

The state after a long time when current and voltage have stabilized and no longer change.

The sum of the voltage drops around any closed loop in a circuit must equal zero. $\sum V = 0$