Problem 8 - Applied Physics

Question

A circuit contains a \(575-\Omega\) resistance, a \(10 \overline{0}-\mu \mathrm{F}\) capacitor, and a \(0.400-\mathrm{H}\) inductance in series with a \(10 \overline{0}-\mathrm{V}, 60 \overline{0}\) - \(\mathrm{Hz}\) source. Find the impedance and the current.

Step-by-Step Solution

Answer

The impedance is approximately \(2096 \,\Omega\) and the current is \(0.0477 \,A\).

Step 1: Identify known quantities

We know the resistance is \( R = 575 \,\Omega \), the capacitance is \( C = 10 \,\mu F = 10 \times 10^{-6} \,F \), the inductance is \( L = 0.400 \,H \), and the source voltage is \( V = 100 \,V \) at a frequency of \( f = 60 \,Hz \).

Step 2: Calculate capacitive reactance

The capacitive reactance is given by \( X_C = \frac{1}{2 \pi f C} \). Substituting the known values, we find:\[ X_C = \frac{1}{2 \times 3.1416 \times 60 \times 10 \times 10^{-6}} = \frac{1}{0.0037699} \approx 2656.49 \,\Omega\]

Step 3: Calculate inductive reactance

The inductive reactance is given by \( X_L = 2 \pi f L \). Substituting the known values, we have:\[ X_L = 2 \times 3.1416 \times 60 \times 0.400 = 150.80 \,\Omega\]

Step 4: Determine the total impedance

The total impedance \( Z \) in an RLC series circuit is given by:\[ Z = \sqrt{R^2 + (X_L - X_C)^2}\]Substituting the calculated values:\[ Z = \sqrt{575^2 + (150.80 - 2656.49)^2} \approx 2095.97 \,\Omega\]

Step 5: Calculate the current through the circuit

Using Ohm's law \( I = \frac{V}{Z} \), the current is:\[ I = \frac{100}{2095.97} \approx 0.0477 \, A\]

Key Concepts

RLC Circuit

An RLC circuit is a type of electrical circuit that includes a Resistor (R), an Inductor (L), and a Capacitor (C) all connected in series or parallel.
In this example, the components are connected in series, which means the current flowing through each component is the same.
These circuits are used across various applications, including radio transmitters and receivers.

• The **resistor** converts the electrical energy into heat, opposing the flow of current.
• The **inductor** stores energy in the form of a magnetic field and opposes changes in current.
• The **capacitor** stores electrical energy in an electric field and opposes changes in voltage.

The unique characteristics of RLC circuits involve their ability to resonate at particular frequencies. This is due to the interplay of resistance, inductance, and capacitance. Understanding the roles each component plays is crucial for analyzing and designing circuits effectively.

Impedance Calculation

Impedance is a fundamental concept in AC circuits that describes how the circuit resists the flow of alternating current. Unlike resistance, impedance accounts for both resistive and reactive characteristics.
In an RLC circuit like the one in our example, the total impedance is affected by the resistance, inductive reactance, and capacitive reactance.

The formula for calculating impedance in a series RLC circuit is:\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]To find the impedance:

• **Calculate Inductive and Capacitive Reactance**: These values determine how much the inductor and capacitor each resist the current.
• **Substitute the Values**: Plug these reactances into the impedance formula alongside the resistance.

The impedance is a complex quantity, representing the overall opposition to alternating current. It involves both magnitude and phase, giving a complete picture of the circuit's behavior.

Reactance

Reactance is the part of impedance that resists the flow of AC current due to capacitance or inductance, but not due to resistance.
It manifests in two forms: **inductive reactance** and **capacitive reactance**.

• **Inductive Reactance (X_L)**: This is related to the inductor and increases with frequency.
  It can be calculated using the formula: \[ X_L = 2 \pi f L \]where \( f \) is the frequency and \( L \) is the inductance.
• **Capacitive Reactance (X_C)**: This is related to the capacitor and decreases with frequency. It is given by: \[ X_C = \frac{1}{2 \pi f C} \]where \( C \) is the capacitance.

Reactance causes the current to lead or lag the applied voltage. Understanding this can help in predicting how the AC circuit will respond at different frequencies, a vital aspect when handling AC circuits.

Ohm's Law

Ohm's Law is a central principle in electronics that relates voltage, current, and resistance in a circuit.
In AC circuits, this law is expanded to incorporate impedance, allowing for the calculation of current when the impedance and voltage are known.

• In its basic form, **Ohm's Law** can be expressed as:\[ I = \frac{V}{R} \]where \( I \) is the current, \( V \) is the voltage, and \( R \) is the resistance.
• For AC circuits, where impedance (\( Z \)) replaces resistance, it is adjusted to:\[ I = \frac{V}{Z} \]This formula takes into account the overall opposition in an AC circuit.

By applying Ohm's Law, one can determine the current flow in a circuit, which is essential for analyzing circuit performance and ensuring proper functionality.