Problem 69 - Physics for Scientists and Engineers with Modern Physics

Question

A ring with a radius of \(3.0 \mathrm{~cm}\) and a resistance of \(0.025 \Omega\) is rotated about an axis through its diameter by \(90^{\circ}\) in a magnetic field of \(0.23 \mathrm{~T}\) perpendicular to that axis. What is the largest number of electrons that would flow past a fixed point in the ring as this process is accomplished?

Step-by-Step Solution

Answer

The largest number of electrons flowing is approximately \(1.62 \times 10^{17}\).

Step 1: Understand the Problem

We have a ring with given dimensions and properties (a radius of \(3.0 \text{ cm}\) and a resistance of \(0.025 \Omega\)). It is being rotated in a magnetic field with a strength of \(0.23 \text{ T}\). We want to determine the maximum number of electrons that pass a point in the ring during this rotation.

Step 2: Calculate the Magnetic Flux Before Rotation

The initial magnetic flux \( \Phi_i \) before rotation can be calculated using the formula \( \Phi = B \cdot A \cdot \cos(\theta) \). Here, \( B = 0.23 \text{ T} \), \( \theta = 0 \text{ degrees} \), and \( A \) is the area of the ring. \( A = \pi r^2 = \pi (0.03 \text{ m})^2 \). Therefore, \( \Phi_i = 0.23 \times \pi \times (0.03)^2 \times \cos(0) \).

Step 3: Calculate the Magnetic Flux After Rotation

The final magnetic flux \( \Phi_f \) after a \(90^{\circ}\) rotation is \( \Phi_f = B \cdot A \cdot \cos(90^{\circ}) = 0 \). This is because the magnetic field is perpendicular now to the area, making the angle \( \theta = 90^{\circ} \), and \( \cos(90^{\circ}) = 0 \).

Step 4: Find the Change in Magnetic Flux

The change in magnetic flux \( \Delta \Phi \) is the difference \( \Phi_i - \Phi_f \). Using the earlier calculated values, \( \Delta \Phi = 0.23 \times \pi \times (0.03)^2 \times 1 - 0 = 0.23 \times \pi \times (0.03)^2 \).

Step 5: Calculate the Induced EMF

The induced electromotive force (EMF) \( \epsilon \) can be found using Faraday's Law: \( \epsilon = -\frac{\Delta \Phi}{\Delta t} \). Assuming an instantaneous rotation, the change in time \( \Delta t \) approaches zero, so primary focus is on \( \Delta \Phi \). The EMF is equal to the product of the resistance and the current (Ohm's Law): \( \epsilon = I \times R \), so \( I = \frac{\epsilon}{R} = \frac{\Delta \Phi}{R \cdot \Delta t} = \frac{0.23 \times \pi \times (0.03)^2}{0.025 \cdot \Delta t} \).

Step 6: Calculate the Charge Passing Through the Ring

The total charge \( q \) passing a point in the ring is \( q = I \times \Delta t \). Combine with the previous step, \( q = \frac{0.23 \times \pi \times (0.03)^2}{0.025} \).

Step 7: Convert Charge to Number of Electrons

We know one electron has a charge of approximately \(1.6 \times 10^{-19} \text{ C}\). The number of electrons \( n \) is \( n = \frac{q}{e} \), where \( q = \frac{0.23 \times \pi \times (0.03)^2}{0.025} \) and \( e = 1.6 \times 10^{-19} \text{ C}\). Calculate \( n = \frac{q}{1.6 \times 10^{-19}} \).

Step 8: Final Calculation and Answer

Calculate \( \Phi_i = 0.23 \times \pi \times (0.03)^2 \approx 0.000648 \text{ Wb}\). Thus \( q = \frac{0.000648}{0.025} = 0.02592 \text{ C} \). Therefore, \( n = \frac{0.02592}{1.6 \times 10^{-19}} \approx 1.62 \times 10^{17} \text{ electrons}\).

Key Concepts

Magnetic Flux

Magnetic flux is a measurement of the total magnetic field passing through a specific area. It is described mathematically by the formula \( \Phi = B \cdot A \cdot \cos(\theta) \), where:

• \( B \) is the magnetic field strength in Tesla (T).
• \( A \) is the area through which the field lines pass, in square meters \( m^2 \).
• \( \theta \) is the angle between the magnetic field direction and the perpendicular to the surface.

When the surface is perpendicular to the magnetic field, \( \theta = 0 \), and \( \cos(0) = 1 \), resulting in maximum magnetic flux. In this exercise, the initial flux was calculated before the ring rotation and after a \(90^{\circ}\) rotation. The difference between these two values is crucial for assessing electromagnetic effects.

Faraday's Law

Faraday's Law of Electromagnetic Induction is a fundamental principle that describes how a change in magnetic flux through a circuit induces an electromotive force (EMF). It can be expressed as \( \epsilon = -\frac{d\Phi}{dt} \).

• \( \epsilon \) represents the induced EMF.
• \( \frac{d\Phi}{dt} \) is the rate of change of magnetic flux \( \Phi \) through the circuit.

The negative sign is Lenz's Law, indicating that the induced EMF generates a current whose magnetic field opposes the change in flux. In the exercise, Faraday's Law is used to calculate the induced EMF when the ring rotates, causing a change from initial to zero magnetic flux.

Number of Electrons

To determine the number of electrons that flow, we first calculate the total charge \( q \) that passes a point due to the induced EMF. The relationship is defined as \( q = I \times \Delta t \), where:

• \( I \) is the induced current.
• \( \Delta t \) is the time interval (approximated as minimal for this exercise).

Once the total charge is calculated, the number of electrons \( n \) is found using \( n = \frac{q}{e} \), with \( e = 1.6 \times 10^{-19} \text{ C} \) being the elementary charge. For this problem, after determining \( q \), converting it using the electron charge yields the desired count of electrons.

Resistance

Resistance in an electrical circuit is the hindrance to the flow of electric charge. It is quantified in Ohms \( \Omega \). The resistance of the ring in the problem affects how much current flows due to the induced EMF.

• Using Ohm's Law, \( V = I \times R \), we see that any induced EMF (\( V = \epsilon \)) in the ring will produce a current \( I \) inversely proportional to its resistance \( R \).
• Therefore, \( I = \frac{\epsilon}{R} \).

In this exercise, the ring's resistance is given as \( 0.025 \Omega \), which is critical in determining the current from the induced EMF, impacting the number of electrons calculated.

Induced EMF

Induced electromotive force (EMF) is the voltage generated by changing the magnetic environment of a circuit. In Faraday's Law, the EMF \( \epsilon \) is induced when there is a change in magnetic flux \( \Delta \Phi \) over time. Specifically, \( \epsilon = -\frac{\Delta \Phi}{\Delta t} \).The induced EMF drives the current through the ring whose magnitude is limited by the resistance of the ring according to Ohm’s Law: \( \epsilon = I \times R \). The exercise assumes near-instantaneous rotation, simplifying \( \Delta t \), and focuses on the flux change which is solely responsible for the EMF's magnitude during this fast process.