Problem 13 - University Physics with Modern Physics

Question

A \(150-\Omega\) resistor is connected in series with a \(0.250-\mathrm{H}\) inductor and an ac source. The voltage across the resistor is \(v_{R}=(3.80 \mathrm{V}) \cos [(720 \mathrm{rad} / \mathrm{s}) t]\) . ( a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage \(v_{L}\) across the inductor.

Step-by-Step Solution

Answer

(a) Current: \(i(t) = 0.0253 \cos(720t) \mathrm{A}\); (b) Inductive reactance: \(180 \Omega\); (c) Inductor voltage: \(v_{L}(t) = 4.554 \cos(720t + \frac{\pi}{2}) \mathrm{V}\).

Step 1: Determine Circuit Current Expression

The voltage across the resistor is given by \(v_{R} = (3.80 \mathrm{V}) \cos [(720 \mathrm{rad} / \mathrm{s}) t]\). The current \(i(t)\) in the circuit can be expressed using Ohm's Law: \(v_{R} = i(t) \times R\), where \(R = 150 \Omega\). Therefore, \(i(t) = \frac{v_{R}}{R} = \frac{3.80}{150} \cos [(720 \mathrm{rad} / \mathrm{s}) t]\). Simplifying, \(i(t) = 0.0253 \cos [(720 \mathrm{rad} / \mathrm{s}) t] \mathrm{A}\).

Step 2: Calculate Inductive Reactance

The inductive reactance \(X_{L}\) of the inductor is calculated using the formula \(X_{L} = \omega L\), where \(\omega = 720 \mathrm{rad/s}\) and \(L = 0.250 \mathrm{H}\). Substituting these values, we get \(X_{L} = 720 \times 0.250 = 180 \Omega\).

Step 3: Derive Expression for Voltage Across Inductor

The voltage across the inductor \(v_{L}\) can be expressed using the formula \(v_{L} = i(t) \times X_{L}\). From Step 1, \(i(t) = 0.0253 \cos [(720 \mathrm{rad} / \mathrm{s}) t] \), and from Step 2, \(X_{L} = 180 \Omega\). Therefore, \(v_{L} = 0.0253 \times 180 \cos [(720 \mathrm{rad} / \mathrm{s}) t + \frac{\pi}{2}] = 4.554 \cos [(720 \mathrm{rad} / \mathrm{s}) t + \frac{\pi}{2}] \mathrm{V}\).

Key Concepts

Inductive Reactance

Inductive reactance is a crucial concept when dealing with alternating current (AC) circuits, especially when inductors are involved. In simple terms, inductive reactance is the opposition that an inductor presents to the change in current. It's like a roadblock for the AC current trying to pass through.

To calculate inductive reactance (\(X_{L}\)), we use the formula: \(X_{L} = \omega L \), where \(\omega = 2 \pi f \) is the angular frequency of the AC source (in rad/s) and \(L\) is the inductance (in Henrys).

In our example, the angular frequency is given as \(720 \text{ rad/s}\) and the inductance \(L = 0.250 \text{ H}\). So, substituting these values into the formula gives us \(X_{L} = 720 \times 0.250 = 180 \: \Omega\). This value tells us how much the inductor resists the AC current, similar to resistance but dependent on frequency.

Ohm's Law

Ohm's Law is a fundamental principle used in electrical engineering to relate voltage, current, and resistance in an electric circuit. It states that the current (\(i\)) through a conductor between two points is directly proportional to the voltage (\(v\)) across the two points and inversely proportional to the resistance (\(R\)). The mathematical formula is \(v = i \times R \).

In the context of AC circuits, Ohm's Law helps us determine the circuit current. From our example, the voltage across the resistor is given by \(v_{R} = (3.80 \text{ V})\cos [(720 \text{ rad/s})t] \). Using Ohm's Law, we rearrange the formula to solve for current: \(i(t) = \frac{v_{R}}{R}\). Substituting \(R = 150 \: \Omega\), the current becomes: \(i(t) = \frac{3.80}{150} \cos [(720 \text{ rad/s}) t] = 0.0253 \cos [(720 \text{ rad/s}) t] \: \text{A}\). This shows how current varies with time in the AC circuit.

Voltage Across Inductor

The voltage across an inductor in an AC circuit can be a bit tricky due to the phase shift involved. As opposed to resistors, inductors cause the current to lag behind the voltage by \(90^\circ \) or \(\frac{\pi}{2} \text{ radians}\).

We calculate the voltage across an inductor (\(v_{L}\)) using the formula: \(v_{L} = i(t) \times X_{L}\). Taking the current expression \(i(t) = 0.0253 \cos [(720 \text{ rad/s}) t]\) and the inductive reactance \(X_{L} = 180 \: \Omega\), we find \(v_{L}\).

Incorporating the phase shift, we derive: \(v_{L} = 0.0253 \times 180 \cos [(720 \text{ rad/s}) t + \frac{\pi}{2}] = 4.554 \cos [(720 \text{ rad/s}) t + \frac{\pi}{2}]\text{ V}\).

This formula illustrates how the voltage across the inductor leads the current due to the introduction of \(\frac{\pi}{2}\) radians, showing the dynamic nature of AC circuits.

Series RLC Circuit

A series RLC circuit consists of a resistor (R), an inductor (L), and a capacitor (C) all connected in a single path. These components interact in a way that affects the total voltage and current in the circuit.

In our example, we focus on the resistor and inductor. In such circuits, the resistance affects how much current can flow, while the inductive reactance influences how the current and voltage phase angle differ.

Overall, the behavior of a series RLC circuit depends on the frequency of the AC source. At a particular frequency called the resonant frequency, the total reactance \(X = X_{L} - X_{C}\) becomes zero, and the impedance is minimum, resulting in maximum current flow. This demonstrates how different components in an RLC circuit balance out their effects based upon frequency.

The example given highlights how a series RLC circuit behaves when only an inductor and resistor are involved, showing the practical application of core circuit concepts like reactance and impedance.