Question

An inductor of inductance \(\mathrm{L}\) and resistor of resistance \(\mathrm{R}\) are joined in series and connected by a source of frequency 0 power dissipated in the circuit is, (a) \(\left[\left\\{\mathrm{R}^{2}+\mathrm{c}^{2} \mathrm{~L}^{2}\right\\} / \mathrm{V}\right]\) (b) \(\left[\left\\{\mathrm{V}^{2} \mathrm{R}\right\\} /\left\\{\mathrm{R}^{2}+\omega^{2} \mathrm{~L}^{2}\right\\}\right]\) (c) \(\left[\mathrm{V} /\left\\{\mathrm{R}^{2}+\omega^{2} \mathrm{~L}^{2}\right\\}\right]\) (d) \(\left[\sqrt{ \left.\left\\{R^{2}+\omega^{2} L^{2}\right\\} / V^{2}\right]}\right.\)

Short answer

The short answer is: \(P = \frac{V^2 R}{R^2 + ω^2 L^2}\)

Step-by-step solution

Determining the Impedance (Z) of the Circuit

For a series R-L circuit, the impedance can be calculated using the formula: \[Z = \sqrt{R^2 + (ωL)^2}\] Where, Z = impedance of the circuit, R = resistance of the resistor, ω = angular frequency, L = inductance of the inductor. Now we can plug in the given values for R, ω, and L to find the impedance Z of this circuit.

Finding the RMS Voltage (V_RMS) and Current (I_RMS)

The power dissipated in the circuit is given by the product of the RMS voltage and RMS current times the power factor (cos θ): \(P = V_{RMS}I_{RMS} \cos \theta\) Where, P = power dissipated, V_RMS = RMS voltage across the circuit, I_RMS = RMS current through the circuit, θ = phase angle between voltage and current, given by: \(\theta = \tan^{-1} \frac{ωL}{R}\), and the power factor is given by: \(\cos \theta = \frac{R}{Z}\)

Calculating the Power Dissipated in Terms of RMS Voltage and Current

To find the power dissipated in the circuit (P), we can rewrite the given equation: \(P = V_{RMS}I_{RMS}\frac{R}{Z}\) Since \(I_{RMS} = \frac{V_{RMS}}{Z}\) We can substitute the current in the equation: \(P = V_{RMS}\left(\frac{V_{RMS}}{Z}\right)\frac{R}{Z}\) Which simplifies to: \(P = \frac{V_{RMS}^2 R}{Z^2}\) Finally, we substitute the impedance formula from step 1: \(P = \frac{V_{RMS}^2 R}{(R^2 + (ωL)^2)}\) Considering the given options: (a) \(\frac{R^2 + ω^2 L^2}{V}\) (b) \(\frac{V^2 R}{R^2 + ω^2 L^2}\) (c) \(\frac{V}{R^2 + ω^2 L^2}\) (d) \(\sqrt{ \frac{R^2 + ω^2 L^2}{V^2}}\) Comparing the final formula with the given options, the correct option is (b): \(P = \frac{V^2 R}{R^2 + ω^2 L^2}\)

Key concepts

Impedance Calculation

In an R-L circuit, the impedance is a crucial concept that represents the total opposition to the flow of alternating current. Both resistance and inductance contribute to this opposition. For a series R-L circuit, the impedance \(Z\) is calculated using the formula: \[Z = \sqrt{R^2 + (\omega L)^2}\] Here, \

• \(R\) stands for the resistance of the resistor.
• \(L\) is the inductance of the inductor.
• \(\omega\) represents the angular frequency, which is related to the frequency of the source.

The value of impedance shows how much the circuit resists the alternating current due to both its resistive and inductive parts. A larger impedance means less current flows through the circuit for a given voltage. Understanding impedance helps in determining how different components affect the circuit's performance.

RMS Voltage and Current

When working with alternating current (AC) circuits like an R-L circuit, RMS (Root Mean Square) values are essential. RMS voltage \(V_{RMS}\) and RMS current \(I_{RMS}\) represent effective or equivalent DC values for power calculations. The RMS values help us to directly use AC voltages and currents in power formulas to evaluate how much power a load consumes or dissipates.Power in an R-L circuit is calculated using these RMS values. The formula is: \[P = V_{RMS}I_{RMS} \cos \theta\]

• \(V_{RMS}\) is the effective AC voltage across the circuit.
• \(I_{RMS}\) is the effective AC current through the circuit.
• \(\cos \theta\) is the power factor, discussing later, representing the efficiency of the circuit in using the energy.

RMS values help in practical sizing and designing of circuit components for safe and efficient energy use in AC systems.

Phase Angle and Power Factor

The phase angle \(\theta\) in an R-L circuit measures the difference in phase between the voltage and current waveforms. It’s a significant aspect of AC analysis. You can calculate this angle using the formula: \[\theta = \tan^{-1} \left(\frac{\omega L}{R}\right)\] In this formula:

• \(\omega L\) is the reactive part due to inductance.
• \(R\) is the resistive part.

The power factor \(\cos \theta\) is crucial because it tells you how effectively a circuit uses power. It’s: \[\cos \theta = \frac{R}{Z}\] Power factor, ranging from 0 to 1, indicates efficiency:

• A power factor close to 1 means most of the power is effectively used, reducing energy losses.
• A lower power factor implies more energy loss as heat.

Understanding phase angle and power factor helps improve circuit efficiency, guiding in the selection of components and adjustments to reduce wasted energy.