Question

A respiration monitor has a flexible loop of copper wire, which wraps about the chest. As the wearer breathes, the radius of the loop of wire increases and decreases. When a person in the Earth's magnetic field (assume \(\left.0.426 \cdot 10^{-4} \mathrm{~T}\right)\) inhales, what is the average current in the loop, assuming that it has a resistance of \(30.0 \Omega\) and increases in radius from \(20.0 \mathrm{~cm}\) to \(25.0 \mathrm{~cm}\) over \(1.00 \mathrm{~s}\) ? Assume that the magnetic field is perpendicular to the plane of the loop.

Short answer

Answer: To find the average current in the loop of copper wire while the person inhales, first, calculate the initial and final magnetic flux for each radius using the magnetic field and the area of the loop. Then, find the change in magnetic flux and divide it by the time taken for the change in radius to determine the induced EMF. Finally, use Ohm's law to find the average current by dividing the induced EMF by the resistance of the loop.

Step-by-step solution

Understand Faraday's law

Faraday's law states that the electromotive force (EMF) induced in a closed loop of wire is equal to the rate of change of the magnetic flux through the loop. Mathematically, this can be written as: EMF = \(-\frac{d\Phi}{dt}\) Where \(\Phi\) is the magnetic flux, which is defined as the product of the magnetic field (B) and the area (A) through which it passes: \(\Phi = B \cdot A\) In this problem, we are given the Earth's magnetic field (B), and the change in the loop's radius (\(\Delta r\)) as the chest expands. Using these variables, we can calculate the change in magnetic flux over a given period of time.

Calculate the initial and final magnetic flux

We are given that the radius of the loop increases from \(20.0 \mathrm{~cm}\) to \(25.0 \mathrm{~cm}\). First, convert the radius values to meters: \(r_1 = 20.0 \mathrm{~cm} \cdot \frac{1}{100 \rm m/cm} = 0.20 \mathrm{~m}\) \(r_2 = 25.0 \mathrm{~cm} \cdot \frac{1}{100 \rm m/cm} = 0.25 \mathrm{~m}\) Since the magnetic field (B) is perpendicular to the plane of the loop, we can use the formula for magnetic flux in a flat loop: \(\Phi_1 = B \cdot A_1 = B \cdot \pi r_1^2\) \(\Phi_2 = B \cdot A_2 = B \cdot \pi r_2^2\) Substitute the given values of B, \(r_1\), and \(r_2\) to find the initial and final magnetic fluxes.

Calculate the change in magnetic flux and the induced EMF

Now, we can find the change in the magnetic flux as the radius of the loop increases: \(\Delta\Phi = \Phi_2 - \Phi_1\) Next, we will find the electromotive force (EMF) by considering the time taken for the radius change: EMF = \(-\frac{d\Phi}{dt} = -\frac{\Delta\Phi}{\Delta t}\) Since we are given that the change in radius occurs over 1.00 second, \(\Delta t = 1.00 \ \mathrm{s}\). Substitute the values of \(\Delta\Phi\) and \(\Delta t\) to find the induced EMF.

Calculate the average current

Now that we have determined the EMF, we can use Ohm's law to find the average current (I) in the loop. Ohm's law is given as: I = \(\frac{EMF}{R}\) Where R is the resistance of the loop. Substitute the given resistance (R) and the calculated EMF to find the average current (I). The resulting current represents the average current in the loop while the person inhales. By following these steps and using the given information, you can find the average current in the loop of copper wire as the person inhales, considering the Earth's magnetic field and the resistance of the loop.

Key concepts

Magnetic Flux

Magnetic flux is a crucial concept in understanding electromagnetic phenomena, particularly when dealing with loops of wire in magnetic fields. It is defined as the product of the magnetic field (\(B\)) and the area (\(A\)) that it penetrates. Mathematically, the magnetic flux (\(\Phi\)) can be expressed as:

1. \(\Phi = B \cdot A\)

This formula suggests that the larger the magnetic field or the area it passes through, the greater the magnetic flux.
In the context of the given problem, the loop's area changes as the wearer breathes, specifically as the radius of the loop changes. The area (\(A\)) for a circular loop is given by \(\pi r^2\). Thus, any change in the radius will affect the magnetic flux.
The initial and final magnetic flux can be determined by using the initial and final radii of the loop, as follows:

• Initial Flux, \(\Phi_1 = B \cdot \pi \cdot r_1^2\)
• Final Flux, \(\Phi_2 = B \cdot \pi \cdot r_2^2\)

Substituting the values of the initial and final radii and the constant magnetic field provided helps us compute these flux values. This change in magnetic flux, important in Faraday's law, is what induces an electromotive force in the loop as breathing occurs.

Ohm's Law

Ohm's Law is fundamental in understanding how currents flow through materials with a given resistance. It provides the relationship between voltage, current, and resistance in a circuit, expressed as:

• I = \(\frac{EMF}{R}\)

Where \(I\) is the current, \(EMF\) is the electromotive force, and \(R\) is the resistance.
In the given respiration monitor problem, after calculating the electromotive force (EMF) using Faraday’s Law, Ohm's Law is applied to find the average current flowing through the loop of copper wire.
The resistance of the loop (\(30.0 \Omega\)) affects how much current flows for a given EMF. Understanding this relationship helps in predicting the behavior of the circuit as the loop's radius changes. Hence, after determining the EMF from the change in magnetic flux over time, utilizing Ohm's Law enables computation of the actual average current:

1. I = \(\frac{EMF}{30.0}\)

Such calculations are essential not just theoretically but also in practical applications like medical monitoring systems.

Electromotive Force (EMF)

Electromotive force, abbreviated as EMF, is a key concept in electromagnetism and plays an essential role in Faraday's Law of Induction. It refers to the voltage developed by any source of electrical energy such as a battery or a changing magnetic field as in the problem.
In Faraday’s Law, the EMF is defined by the rate of change of the magnetic flux through the loop:

1. EMF = \(-\frac{d\Phi}{dt}\)

In the context of a breathing monitor, the EMF is induced in the loop of copper wire as its area changes due to breathing. This change alters the magnetic flux and, over the specific time of 1 second in this case, allows calculating EMF using the formula:

• EMF = \(-\frac{\Delta\Phi}{\Delta t}\)

This induced EMF drives the average current through the loop, as per Ohm’s Law. Understanding how EMF is generated and its implications on current flow is vital in designing and analyzing circuits that involve inductance and changing magnetic fields. Applications include not only respiration monitors but also many devices using electromagnetic induction, such as transformers and electric generators.