Problem 17 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The magnetic field perpendicular to a single 18.2 -cmdiameter circular loop of copper wire decreases uniformly from \(0.750 \mathrm{~T}\) to zero. If the wire is \(2.35 \mathrm{~mm}\) in diameter, how much charge moves past a point in the coil during this operation?

Step-by-Step Solution

Answer

The charge passing a point in the coil is approximately \( Q = \frac{B_i \cdot A}{\rho \cdot 2\pi \cdot 0.091} \). Calculate with given values for exact charge.

Step 1: Determine the change in magnetic flux

The magnetic field initially is \( B_i = 0.750 \; T \) and it drops to \( B_f = 0 \; T \). The area of the loop is calculated as \( A = \pi r^2 \), where the radius \( r = 0.091 \; m \). Thus, \( A = \pi (0.091)^2 \; m^2 \). The initial magnetic flux is \( \Phi_i = B_i \cdot A \) and final magnetic flux is \( \Phi_f = B_f \cdot A = 0 \). The change in flux \( \Delta \Phi \) is \( \Phi_i - \Phi_f = B_i \cdot A \).

Step 2: Calculate the induced electromotive force (EMF)

Using Faraday's law of electromagnetic induction, the magnitude of the induced EMF in the loop is given by \( |\mathcal{E}| = \left| \frac{d\Phi}{dt} \right| \). Since the magnetic field decreases uniformly and the exact time \( t \) is unknown, assume a total change \( \Delta \Phi = B_i \cdot A \).

Step 3: Determine wire properties

The wire diameter is \( 2.35 \; mm \), so the radius \( r_w = 1.175 \times 10^{-3} \; m \), necessary for calculating resistance. Cross-sectional area \( A_w = \pi (1.175 \times 10^{-3})^2 \; m^2 \). The resistance \( R \) of the wire with length \( l \) can be calculated by \( R = \rho \frac{l}{A_w} \), where \( \rho = 1.68 \times 10^{-8} \; \Omega \cdot m \) for copper and \( l = \pi \times 0.182 \; m \) (circumference of the loop).

Step 4: Calculate the charge passing a point in the coil

The induced charge \( Q \) is computed by the formula \( Q = \frac{|\mathcal{E}|}{R} \cdot t \). Given that the EMF is related to change in flux over a period of time, transform it to \( Q = \frac{\Delta \Phi}{R} \). Substituting the resistance and change in flux values will give you the final result.

Key Concepts

Faraday's law of electromagnetic induction

Faraday's law of electromagnetic induction is a fundamental principle that describes how electric currents can be generated by changing magnetic fields. This process is essential in many everyday technologies, such as power generators and electric transformers. The law states that the induced electromotive force (EMF) in a circuit is directly proportional to the rate of change of magnetic flux through the circuit. In other words, a changing magnetic field generates an electric current.

In mathematical terms, Faraday's law can be expressed as:

• \( |\mathcal{E}| = \left| \frac{d\Phi}{dt} \right| \)

Here, \( |\mathcal{E}| \) represents the magnitude of the EMF, and \( \frac{d\Phi}{dt} \) is the rate of change of the magnetic flux \( \Phi \).

Understanding Faraday's law is crucial for solving problems involving electromagnetic induction, like the exercise you encountered. As the magnetic field decreases uniformly, an EMF is induced in the copper loop. This EMF drives a current around the loop, which can be calculated if the rate of change of the magnetic flux is known.

Resistance Calculation

Resistance calculation is an important step in understanding how much current will flow in a circuit when an electromotive force is applied. Resistance is influenced by the material properties and dimensions of the conductor. For a wire, it can be calculated using the formula:

• \( R = \rho \frac{l}{A_w} \)

where \( R \) is the resistance of the wire, \( \rho \) is the resistivity of the material (for copper, \( 1.68 \times 10^{-8} \; \Omega \cdot m \)), \( l \) is the length of the wire, and \( A_w \) is the cross-sectional area of the wire.

In the exercise, the wire's diameter and, consequently, its radius are used to calculate the cross-sectional area \( A_w \). The resistance is then determined using the given formula. The calculation of resistance is necessary for determining how much charge will move in response to an induced EMF, which connects closely to the next concept.

Induced Electromotive Force (EMF)

Induced Electromotive Force, or EMF, is the voltage generated inside a loop or coil when the magnetic environment of the loop is changing. This is directly due to Faraday's law of electromagnetic induction, and it results in the production of current.

In the context of the problem, as the magnetic field decreases from 0.750 T to zero, the change in magnetic flux causes an EMF to be generated in the copper loop. This induced EMF then moves charges through the circuit. Even without a known time duration for the change, the key lies in using the change in flux \( \Delta \Phi \) divided by the calculated resistance \( R \) to determine the charge \( Q \), via the relation:

• \( Q = \frac{\Delta \Phi}{R} \)

Understanding the relationship between EMF and the movement of charge is essential for analyzing circuits influenced by changing magnetic fields.