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Chapter 29: Problem 26

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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