Problem 27 - College Physics

Question

A solenoid has a resistance of \(48.0 \Omega\) and an inductance of \(0.150 \mathrm{H}\). If a \(100 \mathrm{~Hz}\) voltage source is connected across the solenoid, determine the phase angle between the voltage and the current. Does the voltage lead the current or lag the current?

Step-by-Step Solution

Answer

The phase angle is \(63.0^\circ\) and the voltage leads the current.

Step 1: Determine the Inductive Reactance

The inductive reactance (X_L) in an AC circuit is calculated using the formula: \[X_L = 2\pi f L\]where \(f\) is the frequency and \(L\) is the inductance. Substitute \(f = 100\, \text{Hz}\) and \(L = 0.150\, \text{H}\):\[X_L = 2\pi (100)(0.150) = 94.25 \Omega\]

Step 2: Calculate the Total Impedance

The total impedance (Z) of a circuit with a resistor and an inductor is calculated by:\[Z = \sqrt{R^2 + X_L^2}\]Substitute \(R = 48.0\, \Omega\) and \(X_L = 94.25\, \Omega\):\[Z = \sqrt{(48.0)^2 + (94.25)^2} = 106.3 \Omega\]

Step 3: Calculate the Phase Angle

The phase angle (\phi) between the voltage and the current is given by:\[\phi = \arctan\left(\frac{X_L}{R}\right)\]Substitute \(X_L = 94.25\, \Omega\) and \(R = 48.0\, \Omega\):\[\phi = \arctan\left(\frac{94.25}{48.0}\right) = 1.10 \text{ radians} = 63.0^\circ\]

Step 4: Determine the Lead-Lag Relationship

In an inductive circuit, the voltage leads the current by the phase angle calculated. Since we have a positive phase angle, the voltage leads the current.

Key Concepts

Inductive Reactance

Inductive reactance is a crucial concept when dealing with AC circuits, especially those involving inductors like solenoids. It represents the opposition that an inductor presents to a change in current. Unlike resistance, which opposes both AC and DC current, inductive reactance specifically affects AC current. It is determined using the formula:

• \(X_L = 2\pi f L\)

where \(f\) is the frequency of the AC source and \(L\) is the inductance.
The higher the frequency or inductance, the greater the opposition to current flow. This property makes inductors particularly effective at blocking high-frequency signals while allowing DC or low-frequency signals to pass.
For example, in our problem, with a frequency of 100 Hz and an inductance of 0.150 H, the inductive reactance is calculated as 94.25 Ω. This value is significant as it partially determines how the circuit will behave.

Phase Angle

The phase angle in an AC circuit indicates the timing difference between the voltage and current waveforms. Calculating the phase angle helps us understand whether the voltage leads or lags the current.

• It is found using: \(\phi = \arctan\left(\frac{X_L}{R}\right)\)

For our example, with an inductive reactance of \(94.25 \Omega\) and resistance of \(48.0 \Omega\), the phase angle is calculated to be 63.0°, indicating that voltage leads current significantly. This behavior is typical in inductive circuits, where the stored magnetic energy causes the current to lag behind the voltage. A positive phase angle means that the voltage reaches its maximum before the current does.

Voltage and Current Relationship

Understanding the relationship between voltage and current in AC circuits is essential. Unlike DC circuits, where voltage and current are in phase, AC circuits with inductors will often see a phase difference. In inductive circuits, such as the one in our exercise, the voltage generally leads the current. This is due to the inductor storing energy in a magnetic field and releasing it back into the circuit. Therefore, current lags behind because it takes time to build the magnetic field as current increases.
This relationship is key to calculating other circuit parameters and is especially important in designing AC systems, ensuring they function as expected across different frequency ranges.

Total Impedance

Total impedance is like the "total resistance" in a circuit with both resistors and inductors, though it applies to AC circuits. It combines resistance and inductive reactance into a single value, represented as \(Z\).

• The formula used: \(Z = \sqrt{R^2 + X_L^2}\)

The impedance of a circuit determines how much it opposes the overall flow of AC current.
In our example, with \(R = 48.0 \Omega\) and \(X_L = 94.25 \Omega\), the total impedance calculated is \(106.3 \Omega\). This total impedance affects how much current will flow for a given voltage. Understanding total impedance is fundamental for analyzing and predicting circuit behavior, particularly when designing circuits to operate at different frequencies.