Question

A toroidal solenoid has 2900 closely wound turns, cross-sectional area 0.450 cm\(^2\), mean radius 9.00 cm, and resistance \(R\) = 2.80 \(\Omega\). Ignore the variation of the magnetic field across the cross section of the solenoid. What is the amplitude of the current in the solenoid if it is connected to an ac source that has voltage amplitude 24.0 V and frequency 495 Hz?

Short answer

The amplitude of the current is calculated by finding the impedance and using Ohm's Law for AC: \(I = \frac{24.0}{Z}\).

Step-by-step solution

Convert Cross-Sectional Area to Meters Squared

The cross-sectional area is given in cm², and for solving this problem, we need it in m². Therefore, we need to convert 0.450 cm² to m². We do this by using the conversion factor: 1 cm² = 0.0001 m². \[ 0.450 \text{ cm}^2 = 0.450 \times 0.0001 \text{ m}^2 = 4.50 \times 10^{-5} \text{ m}^2 \]

Calculate the Inductance (L) of the Solenoid

The inductance of a toroidal solenoid can be calculated using the formula: \[ L = \mu_0 \cdot \frac{N^2 \cdot A}{2\pi \cdot r} \] where \(\mu_0 = 4\pi \times 10^{-7} \text{ T m/A} \), \(N = 2900\) turns, \(A = 4.50 \times 10^{-5} \text{ m}^2 \), and \(r = 9.00 \text{ cm} = 0.09 \text{ m}\). Substituting these values in, we get: \[ L = 4\pi \times 10^{-7} \cdot \frac{2900^2 \cdot 4.50 \times 10^{-5}}{2\pi \cdot 0.09} \]

Evaluate the Inductive Reactance (X_L)

With the inductance known, we calculate the inductive reactance \(X_L\) using the formula: \[ X_L = 2\pi f L \] where \(f = 495\text{ Hz}\). We substitute \(L\) from Step 2 and the frequency to find the value of \(X_L\).

Calculate the Impedance (Z) of the Circuit

Using the resistance \(R = 2.80 \Omega\) from the problem and the inductive reactance \(X_L\) calculated in Step 3, compute the total impedance using: \[ Z = \sqrt{R^2 + X_L^2} \]

Determine the Amplitude of the Current

The amplitude of the current \(I\) can be determined using Ohm's Law for an AC circuit: \[ I = \frac{V}{Z} \] where \(V\) is the given voltage amplitude, 24.0 V.

Key concepts

Inductance Calculation

Inductance is a critical concept when working with circuits that include coils such as solenoids. In a toroidal solenoid, which is essentially a coil shaped like a donut, the inductance can be calculated using a specific formula. This gives us a measure of how effectively the solenoid can generate a magnetic field with a certain current. The formula for the inductance, denoted as \( L \), of a toroidal solenoid is: \[ L = \mu_0 \cdot \frac{N^2 \cdot A}{2\pi \cdot r} \]Here:

• \( \mu_0 \) is the permeability of free space, which is a constant \( 4\pi \times 10^{-7} \text{ T m/A} \).
• \( N \) is the total number of turns in the solenoid.
• \( A \) is the cross-sectional area in square meters.
• \( r \) is the mean radius of the solenoid in meters.

When calculating inductance, ensure all units are in the International System of Units (SI). This guarantees accuracy in your calculations. For instance, converting the area from cm² to m² is essential in this calculation. Inductance quantifies how a solenoid opposes changes in current, owing to its electromagnetic properties.

AC Circuit Analysis

AC circuit analysis involves understanding how alternating current behaves in different components of an electrical circuit. It's different from DC circuits, where the current flows only in one direction. In AC circuits, the current reverses direction periodically. This behavior is described by the frequency of the AC supply. When analyzing AC circuits that incorporate inductive components, like our toroidal solenoid, it is important to consider how inductance introduces a phase difference between the voltage and current. It suggests that the voltage leads the current by a phase angle, which is important for calculating other parameters like inductive reactance and impedance. In this exercise, the AC source voltage amplitude is given as 24.0 V with a frequency of 495 Hz. This forms a critical aspect of problem-solving in AC circuit contexts, as both parameters directly influence how energy is transferred and used within the circuit components.

Inductive Reactance

Inductive reactance is a measure of how much a solenoid or inductor resists the flow of alternating current. It is an essential variable when calculating the impedance of circuits with inductors. The reactance, denoted by \( X_L \), depicts the dependency on the frequency of the AC current and the inductance of the solenoid.The formula for inductive reactance is given by:\[ X_L = 2\pi f L \]where:

• \( f \) is the frequency of the AC source in hertz (Hz).
• \( L \) is the inductance in henrys (H).

A higher frequency or greater inductance leads to a larger inductive reactance, indicating a higher opposition to the flow of AC current. This reactance contributes to the circuit's overall impedance, affecting the current flow within the system. Understanding this concept is fundamental, specifically when designing or analyzing AC circuits.

Impedance Calculation

Impedance is akin to resistance but specifically applies to AC circuits. It's a combination of the circuit's resistance and its reactance, which includes both inductive and capacitive reactance. For a circuit with resistance and inductive reactance, the impedance, represented as \( Z \), can be calculated using the formula:\[ Z = \sqrt{R^2 + X_L^2} \]where:

• \( R \) is the resistance in ohms (Ω).
• \( X_L \) is the inductive reactance calculated previously.

The impedance provides a complete picture of how much the circuit opposes the flow of AC current. This value influences the current magnitude according to Ohm's Law, stated in the context of AC circuits as \( I = \frac{V}{Z} \). Understanding and calculating impedance is crucial for correctly predicting the behavior of AC circuits, which is necessary for tasks such as designing circuits and troubleshooting electrical systems.