Problem 19 - Physics

Question

Multiple-Concept Example 3 reviews some of the basic ideas that are pertinent to this problem. A circuit consists of a \(215-\Omega\) resistor and a 0.200 -H inductor. These two elements are connected in series across a generator that has a frequency of \(106 \mathrm{~Hz}\) and a voltage of \(234 \mathrm{~V}\). (a) What is the current in the circuit? (b) Determine the phase angle between the current and the voltage of the generator.

Step-by-Step Solution

Answer

(a) 0.925 A, (b) 31.98°

Step 1: Calculate the Inductive Reactance

First, we need to calculate the inductive reactance (\(X_L\)) of the circuit. The inductive reactance is given by the formula \(X_L = 2\pi f L\), where \(f\) is the frequency and \(L\) is the inductance of the inductor.Using \(f = 106 \text{ Hz}\) and \(L = 0.200 \text{ H}\), we have:\[X_L = 2 \pi (106) (0.200) = 133.71 \Omega\]

Step 2: Determine the Total Impedance

The impedance (\(Z\)) of an R-L circuit is found using the formula \(Z = \sqrt{R^2 + X_L^2}\), where \(R\) is the resistance and \(X_L\) is the inductive reactance.Using \(R = 215 \Omega\) and \(X_L = 133.71 \Omega\):\[Z = \sqrt{(215)^2 + (133.71)^2} = \sqrt{46225 + 17877.1} = 253 \Omega\]

Step 3: Calculate the Current in the Circuit

The current (\(I\)) through the circuit can be obtained using Ohm’s Law, \(I = \frac{V}{Z}\), where \(V\) is the voltage across the generator and \(Z\) is the impedance.Given \(V = 234 \text{ V}\) and \(Z = 253 \Omega\), we have:\[I = \frac{234}{253} = 0.925 \text{ A}\]

Step 4: Calculate the Phase Angle

The phase angle (\(\phi\)) between the current and voltage in an R-L circuit is given by \(\tan \phi = \frac{X_L}{R}\).Using \(X_L = 133.71 \Omega\) and \(R = 215 \Omega\):\[\tan \phi = \frac{133.71}{215} = 0.6228\]To find \(\phi\), calculate the arctangent:\[\phi = \arctan(0.6228) = 31.98^{\circ}\]

Key Concepts

Inductive Reactance in Circuits

When analyzing an R-L Circuit (Resistor-Inductor Circuit), understanding inductive reactance is essential. Inductive reactance, often denoted as \(X_L\), arises from the inductor's opposition to changes in current. This is particularly significant in AC circuits, where the current changes direction and magnitude continuously. The formula for inductive reactance is \(X_L = 2\pi f L\). Here, \(f\) represents the frequency (in Hertz), and \(L\) is the inductance (in Henrys) of the inductor. As the frequency increases, the inductive reactance grows stronger. This means that higher frequencies lead to higher opposition to current flow. - Calculate \(X_L\) using given values of inductance and frequency. - It affects how easily current can flow through the circuit. Understanding this concept helps in determining how an inductor will behave in a given circuit configuration and its impact on the overall circuit behavior.

Applying Ohm's Law in R-L Circuits

Ohm's Law is a fundamental principle used in understanding electrical circuits. Particularly in R-L circuits, Ohm’s Law helps to determine how the voltage, current, and impedance relate. The basic formula is \(I = \frac{V}{Z}\), where \(I\) is the current, \(V\) is the voltage, and \(Z\) is the impedance. Applying this to the circuit, we can understand how much current is flowing when the total impedance and applied voltage are known. - Use the formula to find the current: given voltage and calculated impedance. - It simplifies the step to find the current in complex circuits by understanding the overall resistance (impedance) the system offers. Ohm's Law is indispensable for quickly determining current, provided you have voltage and impedance values for the circuit.

Understanding Phase Angle Calculation

Phase angle in an R-L circuit indicates the difference in phase between the voltage and current waveforms. It is expressed in degrees and provides insights into how the reactive component (inductor) affects the current phase relative to the voltage. The tangent of the phase angle \(\phi\) is calculated using \(\tan \phi = \frac{X_L}{R}\), where \(X_L\) is the inductive reactance and \(R\) is the resistance. Calculating the arctangent of that ratio gives the phase angle in degrees. - Phase angle determines lag/lead behavior in AC circuits. - In R-L circuits, current lags behind voltage due to the inductor. Understanding phase angle is vital for applications requiring synchronization of phase, such as in AC power systems and signal processing.

Comprehending Circuit Impedance in R-L Circuits

Circuit impedance ( Z ) is a critical factor in R-L circuits as it represents the total opposition to alternating current. Impedance combines resistance ( R ) and inductive reactance ( X_L ) into a single complex number. It is calculated as \(Z = \sqrt{R^2 + X_L^2}\). This determines how much the circuit resists the flow of electric current. - Impedance reflects both resistive and reactive elements. - Affects current flow based on circuit configuration and frequency. Understanding impedance in R-L circuits is crucial for designing and analyzing circuits in electronics, especially those involving AC where both resistance and reactance are present.