RL Circuits Comprehensive Review

Definition of an RL Circuit: An RL circuit is an electrical circuit that contains both a resistor ( $R$ ) and an inductor ( $L$ ). These circuits are analyzed similarly to RC circuits (resistor-capacitor), focusing on the time-dependent behavior of the current.
The Role of the Resistor: The resistor provides resistance to the flow of current. Its behavior is governed by Ohm's Law ( $V = IR$ ).
The Role of the Inductor: The inductor provides resistance specifically to the change in current ( $\frac{\Delta I}{\Delta t}$ ) rather than the flow itself. It functions via self-inductance.
Self-Inductance and Back EMF: When current in a loop changes, it creates a back EMF (electromotive force) that opposes that change. Consequently, the current in an RL circuit does not reach its maximum value instantaneously; it takes a finite amount of time to reach the value predicted by Ohm's Law.
Circuit Simplification Assumptions:
- When a specific inductor component is present in a circuit, any incidental self-inductance caused by the shape of the wire loops is typically ignored.
- Similarly, when specific resistors are present, the resistance of the connecting wires is assumed to be negligible ( $R_{\text{wires}} \approx 0$ ).

Current Growth Equation: As current increases in an RL circuit after a switch is closed, its value at time $t$ is given by the formula: $I(t) = I_{\text{max}} (1 - e^{-\frac{t}{\tau}})$ Where $I_{\text{max}} = \frac{\Delta V}{R}$ .
Asymptotic Behavior:
- At the starting time ( $t = 0$ ), the current $I$ is zero.
- At very large time intervals ( $t \rightarrow \infty$ ), the term $e^{-\frac{t}{\tau}}$ approaches zero, and the current reaches its maximum value, $I_{\text{max}}$ , as predicted by Ohm's Law ( $I = \frac{\Delta V}{R}$ ).
The Five-Time-Constant Rule: In practical application, once five time constants ( $5\tau$ ) have passed, the current is considered to have reached its maximum value. At this point, the current is approximately $99.3\%$ of its maximum.

Formula for the RL Time Constant: The time constant ( $\tau$ ) represents the characteristic time scale for changes in the circuit: $\tau = \frac{L}{R}$
Units and Dimensional Analysis:
- In SI units, $\tau$ is measured in seconds ( $s$ ).
- The unit for inductance is the Henry ( $H$ ), which is equivalent to a volt-second per ampere ( $V \cdot s \cdot A^{-1}$ ).
- The unit for resistance is the Ohm ( $\Omega$ ), equivalent to a volt per ampere ( $V \cdot A^{-1}$ ).
- Dividing $\frac{V \cdot s}{A}$ by $\frac{V}{A}$ results in seconds ( $s$ ). This confirms that the exponent unit ( $\frac{t}{\tau}$ ) is unitless.

Slope and Rate of Change: In a graph of current ( $I$ ) versus time ( $t$ ), the current starts at zero and curves upward toward the maximum. The slope of the line ( $\frac{\Delta I}{\Delta t}$ ) represents the rate of change of current.
- Maximum Slope: Occurs at the very beginning ( $t = 0$ ).
- Minimum Slope: Approaches zero as current levels off at its maximum.
Potential Drop Across the Inductor: The potential difference across an inductor is defined as: $\Delta V_L = -L \frac{\Delta I}{\Delta t}$
- Initial State: Because the rate of change ( $\frac{\Delta I}{\Delta t}$ ) is greatest at the instant the switch is closed, the inductor provides the greatest resistance and has the largest potential drop at that moment.
- Steady State: Once max current is reached, $\frac{\Delta I}{\Delta t} = 0$ , meaning there is no potential drop across the inductor.
Kirchhoff’s Loop Rule: In a single-loop circuit, the sum of potential increases (from sources like batteries) must be balanced by potential decreases (across resistors and inductors). $\sum \Delta V = 0$

Magnetic Field Storage: An inductor, usually a coil of wire or a solenoid, stores energy within its magnetic field when current flows through it.
Energy Formula: The potential energy ( $U_{L}$ ) stored in an inductor is: $U_L = \frac{1}{2} L I^2$ Where $L$ is the inductance and $I$ is the current.
Comparison to Capacitors: This formula follows a similar format to the energy stored in a capacitor ( $U_C = \frac{1}{2} C (\Delta V)^2$ ).
Energy Fluctuations:
- Initial energy is zero because the current is zero.
- Maximum energy is reached when the current is at its maximum ( $I_{\text{max}}$ ).

Scenario Parameters:

Part 1: Current After a Very Long Time

Reasoning: The inductor acts as a simple wire with no potential drop once current is steady.
Calculation: $I = \frac{\Delta V}{R} = \frac{12\,V}{4.0\,\Omega} = 3.0\,A$

Part 2: Potential Drop Across Resistor Immediately After Closing Switch

Part 3: Potential Drop Across Inductor Immediately After Closing Switch

Reasoning: By Kirchhoff's Loop Rule, the sum of drops must equal the source. Since the resistor drop is $0\,V$ , all $12\,V$ must drop across the inductor.
Calculation: $\Delta V_L = -12\,V$

Part 4: Current Passing Through Resistor at $2\,ms$

Step A: Calculate the Time Constant ( $\tau$ ): $\tau = \frac{L}{R} = \frac{0.0080\,H}{4.0\,\Omega} = 0.002\,s = 2\,ms$
Step B: Use the Time Dependence Equation: At $t = 2\,ms$ (which is exactly $1\tau$ ): $I(2\,ms) = 3.0\,A \times (1 - e^{-\frac{2\,ms}{2\,ms}}) = 3.0\,A \times (1 - e^{-1})$
Numerical Result: $I \approx 3.0\,A \times (1 - 0.368) \approx 3.0\,A \times 0.632 = 1.896\,A$
Final Answer: $1.9\,A$ (rounded to two significant figures).

Part 5: Potential Energy Stored After a Very Long Time

Calculation: $U_L = \frac{1}{2} L I_{\text{max}}^2 = \frac{1}{2} (0.0080\,H) (3.0\,A)^2$
Computation: $0.5 \times 0.0080 \times 9 = 0.036\,J$
Final Answer: $0.036\,Joules$