Question

Consider an \(\mathrm{RL}\) circuit with resistance \(R=1.00 \mathrm{M} \Omega\) and inductance \(L=1.00 \mathrm{H}\), which is powered by a \(10.0-\mathrm{V}\) battery. a) What is the time constant of the circuit? b) If the switch is closed at time \(t=0\), what is the current just after that time? After \(2.00 \mu \mathrm{s}\) ? When a long time has passed?

Short answer

Answer: The time constant of the circuit is 1 s. The current just after the switch is closed is 10 μA, after 2 μs it is approximately 9.998 μA, and when a long time has passed, it is 10 μA.

Step-by-step solution

Find the time constant of the circuit

To find the time constant, use the formula τ = L/R, where L is the inductance and R is the resistance. In this case, L = 1 H and R = 1 MΩ, so the time constant is: τ = L/R = (1 H)/(1 MΩ) = 1 s.

Find the current just after the switch is closed

When the switch is closed at t = 0, we have: I(0) = V/R * (1 - e^(0/τ)) = (10 V)/(1 MΩ) * (1 - e^0) = 10 μA.

Find the current after 2 microseconds

We want to find the current at t = 2 μs, so we plug this into the current formula: I(2 μs) = V/R * (1 - e^(-t/τ)) = (10 V)/(1 MΩ) * (1 - e^(-2 μs/1 s)) ≈ 9.998 μA.

Find the current when a long time has passed

When a long time has passed, the inductor acts as a short circuit, and the current reaches its maximum value as given by Ohm's Law: I(max) = V/R = (10 V)/(1 MΩ) = 10 μA. In summary, the time constant of the circuit is 1 s, the current just after the switch is closed is 10 μA, the current after 2 μs is approximately 9.998 μA, and the current when a long time has passed is 10 μA.

Key concepts

Electrical Resistance

Understanding electrical resistance is crucial for comprehending how an RL circuit behaves. Resistance, denoted by the symbol 'R', is a measure of the opposition to the flow of electric current through a conductor. It is comparable to friction in mechanical systems and determines how much current will flow for a given voltage, based on Ohm's Law, which states that the current (I) is equal to the voltage (V) divided by the resistance (R), or, in equation form, \( I = \frac{V}{R} \).

Higher resistance means less current can flow, while lower resistance allows more current. The units of electrical resistance are ohms (Ω). In the exercise provided, the circuit has a resistance of 1 megaohm (1 MΩ) which significantly restricts the current. The concept of electrical resistance is fundamental in predicting current flow and in designing circuits that meet specific requirements for electrical devices.

Inductor in Circuits

An inductor is a passive electrical component that stores energy in a magnetic field when an electric current flows through it. It is typically a coil of wire, and its ability to resist changes in the current is known as inductance, represented by 'L'. This property makes inductors a key element in various types of circuits, especially in filtering and timing applications.

In the context of an RL circuit, which includes a resistor (R) and an inductor (L) in series, the inductor contributes to the circuit's transient response — that is, how the circuit reacts to changes over time, such as when a circuit is switched on or off. The inductor in the given exercise has an inductance of 1 henry (H), which indicates the amount of magnetic flux generated for each ampere of current that flows through the coil.

Transient Response in RL Circuits

The transient response in RL circuits refers to how the circuit reacts to a change, such as suddenly being powered on or off. An important characteristic of this response is the 'time constant' (τ), which signifies how quickly the current in an RL circuit will rise or fall when a voltage is applied or removed. The time constant is a product of the circuit's resistance and inductance, calculated by τ = L/R.

In our example, with an inductance of 1 H and resistance of 1 MΩ, the time constant is 1 second. This means that the circuit will take approximately 1 second to reach around 63% of its final current value after the power is applied. The transient response is crucial for timing and shaping signals in electronic systems. Understanding the time constant helps in predicting the behavior of the circuit over time, and can be utilized for tasks such as signal processing and the timing of events within an electronic system.