Question

What is the inductance in a series \(\mathrm{RL}\) circuit in which \(R=3.00 \mathrm{k} \Omega\) if the current increases to one half of its final value in \(20.0 \mu \mathrm{s} ?\)

Short answer

Answer: The inductance of the series RL circuit is approximately \(4.16 \times 10^{-5} H\).

Step-by-step solution

Calculate the Time Constant (τ)

We are given that the current reaches half of its final value in \(20.0 \mu s\). According to the analysis, the time constant (τ) of the RL circuit can be calculated with the formula: \(I(t) = I_{max}(1 - e^{-t/\tau})\). To reach half of the final current, we set \(I(t) = 0.5I_{max}\). Then, we can solve for τ: \(0.5I_{max} = I_{max}(1 - e^{-t/\tau})\) Divide both sides by \(I_{max}\): \(0.5 = 1 - e^{-t/\tau}\) We are given that the time taken (t) is \(20.0 \mu s\). Now, we need to solve for τ: \(0.5 = 1 - e^{-20\times10^{-6}/\tau}\) Now, subtract 1 from both sides and multiply by -1: \(0.5 = e^{-20\times10^{-6}/\tau}\) Take the natural logarithm of both sides: \(ln(0.5) = -20\times10^{-6}/\tau\) Solve for τ: \(\tau = -\frac{20\times10^{-6}}{ln(0.5)}\)

Solve for Inductance (L)

Now that we have the time constant (τ), we can use the formula τ = L/R to solve for the inductance (L). Rearrange the formula to solve for L: \( L = \tau \times R \) We are given that the resistance (R) is \(3.00 k\Omega = 3000\Omega\). Plug in the values of τ and R into the formula: \(L = \left(-\frac{20\times10^{-6}}{ln(0.5)}\right) \times 3000\) Calculate the inductance (L): \(L \approx 4.16 \times 10^{-5} H\)

Final Answer

The inductance in the series RL circuit is approximately \(4.16 \times 10^{-5} H\).

Key concepts

Inductance

Inductance is a property of an electrical circuit or component that causes it to oppose a change in the current flowing through it. In the case of an RL circuit, which consists of a resistor (R) and an inductor (L) connected in series, inductance plays a critical role. The inductor stores energy in the form of a magnetic field when current passes through it. This energy storage capability leads to the inductor's ability to resist changes in current. The unit of inductance is henry (H), and it is represented by the symbol 'L'. Inductance is crucial because it determines how quickly the current can change in a circuit. In other words, a higher inductance means that it takes more time for the current to reach its new value after a change. This is why inductance is a key value when analyzing RL circuits, especially when looking at time-dependent behaviors.

Time Constant

In an RL circuit, the time constant (\(\tau\)) is an essential concept that describes how fast the circuit responds to changes in voltage or current. The time constant is the time it takes for the current or voltage to rise to about 63.2% of its final value after a sudden change. Similarly, it's the time it takes for it to decay to approximately 36.8% if the initial action is removed. The time constant is calculated as:
\[\tau = \frac{L}{R}\]
where \(L\) is the inductance and \(R\) is the resistance.

• This formula indicates that a larger inductance or a smaller resistance results in a longer time constant.
• The time constant reflects how "fast" or "slow" the circuit is in responding to changes. A long time constant means a slow response, which is often desired in certain applications like filtering.

Understanding the time constant helps in designing circuits for specific responses to changes, whether in communication systems or power electronics.

Natural Logarithm

The natural logarithm, denoted as \(ln\), is a useful mathematical function in science and engineering. It is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.718. Natural logarithms are helpful in solving exponential decay problems, such as those found in RL circuits. In the exercise, it’s used to solve for the time constant (\(\tau\)) when the current reaches a specific fraction of its final value.

• When you have an equation like \(0.5 = e^{-t/\tau}\), taking the natural logarithm of both sides helps to isolate the variable \(\tau\).
• \(ln(0.5)\) represents the transformation from an exponential equation into a linear one, making it much easier to solve.

The concept of natural logarithms is essential in characterizing how RL circuits behave over time, allowing us to model the rate of change in circuits accurately. The use of \(ln\) simplifies complex exponential functions into more manageable algebraic equations, a key technique in both theoretical and applied electrical engineering.