Problem 42 - University Physics with Modern Physics

Question

A toroidal solenoid has 2900 closely wound turns, cross-sectional area \(0.450 \mathrm{cm}^{2},\) mean radius \(9.00 \mathrm{cm},\) and resistance \(R=2.80 \Omega .\) The variation of the magnetic field across the cross section of the solenoid can be neglected. What is the amplitude of the current in the solenoid if it is connected to an ac source that has voltage amplitude 24.0 \(\mathrm{V}\) and frequency 365 \(\mathrm{Hz}\) ?

Step-by-Step Solution

Answer

The amplitude of the current in the solenoid is approximately 8.50 A.

Step 1: Understand the AC Circuit

In an AC circuit with a solenoid, the impedance is a combination of resistance (R) and inductive reactance (X_L). The formula for the impedance (Z) in an RL circuit is: \[ Z = \sqrt{R^2 + X_L^2} \] where \[ X_L = 2\pi f L \] and \( f \) is the frequency of the AC source, and \( L \) is the inductance of the solenoid.

Step 2: Calculate the Inductance of the Solenoid

The inductance \( L \) of a toroidal solenoid can be calculated using the formula: \[ L = \frac{\mu_0 N^2 A}{2\pi r} \] where \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \, \text{Tm/A}) \), \( N \) is the number of turns, \( A \) is the cross-sectional area, and \( r \) is the mean radius.Given: \( N = 2900 \), \( A = 0.450 \times 10^{-4} \, \text{m}^2 \), \( r = 0.09 \, \text{m} \).Substituting the values, we calculate the inductance \( L \).

Step 3: Calculate Inductive Reactance

Using the inductance calculated in Step 2, find the inductive reactance \( X_L \) using: \[ X_L = 2\pi f L \] where \( f = 365 \, \text{Hz} \).

Step 4: Calculate the Impedance

Substitute the resistance \( R = 2.80 \, \Omega \) and the inductive reactance \( X_L \) from Step 3 into the impedance formula: \[ Z = \sqrt{R^2 + X_L^2} \] to calculate the total impedance \( Z \).

Step 5: Calculate Current Amplitude

Use Ohm's Law for AC circuits to find the amplitude of the current \( I \): \[ I = \frac{V}{Z} \] where \( V = 24.0 \, \text{V} \) is the voltage amplitude.

Key Concepts

AC Circuits

An AC circuit is one where the current alternates direction and magnitude. This is common in household electrical systems, which typically have a sinusoidal wave form. An AC source supplies alternating voltage and current, meaning its value changes over time. The frequency of this change is measured in Hertz (Hz), which is the number of cycles per second.
In an AC circuit involving a solenoid, like the toroidal solenoid in our problem, we need to consider more than just the resistance. Because the electric field is constantly changing, it affects the magnetic field in the solenoid. It induces an additional property called inductive reactance, which opposes the change in current and adds to the total impedance of the circuit.

Inductance Calculation

Inductance is a property of an electrical conductor that opposes the change in current. In a toroidal solenoid, the inductance can be calculated using specific parameters. The formula
\[ L = \frac{\mu_0 N^2 A}{2\pi r} \]
lets us calculate inductance, where:

• \( \mu_0 \) is the permeability of free space.
• \( N \) is the number of turns.
• \( A \) is the cross-sectional area.
• \( r \) is the mean radius.

In the given exercise,

• \( N = 2900 \)
• \( A = 0.450 \times 10^{-4} \, \text{m}^2 \)
• \( r = 0.09 \, \text{m} \)

Substituting the values, we can find the inductance \( L \) of the solenoid, which is crucial for determining how the circuit responds to an AC electrical source.

Impedance in RL Circuit

When dealing with an RL (Resistor-Inductor) circuit in the context of AC circuits, total impedance must be calculated. The impedance, represented by \( Z \), is a combination of the resistance \( R \) and the inductive reactance \( X_L \). The formula to find impedance in such a circuit is:
\[ Z = \sqrt{R^2 + X_L^2} \]
Here, the resistance \( R \) was given as \( 2.80 \, \Omega \), and the inductive reactance \( X_L \) can be determined using the formula \( X_L = 2\pi f L \).
It is essential as impedance impacts how much current will flow through the circuit for a given voltage. By combining \( R \) and \( X_L \) using the formula, the complete opposition to current within the circuit can be understood, which directly influences the current flowing through.

Inductive Reactance

Inductive reactance is a crucial factor in AC circuits, especially those containing inductors like our toroidal solenoid. Inductive reactance \( X_L \) specifically measures how much the inductor opposes the AC current's change. It is dependent on the frequency of the AC battery and the inductor's inductance.
The formula \( X_L = 2\pi f L \) helps calculate the inductive reactance, where \( f \) is the frequency of the AC source and \( L \) is the inductance. In the problem, the frequency \( f \) is given as \( 365 \, \text{Hz} \).
By calculating \( X_L \), we understand how the inductor affects the circuit by resisting the change in current. This is why even a circuit with low resistance can have significant impedance if the inductive reactance is high, affecting the circuit's current flow.