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 \(0.426 \cdot 10^{-4} \mathrm{~T}\) ) 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 s? Assume that the magnetic field is perpendicular to the plane of the loop.

Short answer

Using Faraday's Law of electromagnetic induction, we find the average current in a flexible loop of copper wire as the wearer inhales by following these steps: 1. Calculate the change in magnetic flux: \(\Delta \Phi = (0.426 \cdot 10^{-4} \mathrm{~T}) (\pi (25.0 \mathrm{~cm})^2 - \pi (20.0 \mathrm{~cm})^2)\) 2. Calculate the induced EMF: \(\epsilon = -\frac{\Delta \Phi}{1.00 \mathrm{~s}}\) 3. Calculate the average current: \(I = \frac{\epsilon}{30.0 \Omega}\) The average current is found to be \(2.81 \times 10^{-6} A\), or \(2.81 \mathrm{~\mu A}\).

Step-by-step solution

Calculate the change in magnetic flux

To find the induced EMF, we first need to calculate the change in magnetic flux through the loop. The magnetic flux \(\Phi\) through a loop is given by: \(\Phi = B \cdot A \cdot \cos{\theta}\) where \(B\) is the magnetic field, \(A\) is the area of the loop, and \(\theta\) is the angle between the magnetic field and the normal to the loop's plane. In our case, the magnetic field is perpendicular to the plane of the loop, so \(\theta = 90^\circ\) and \(\cos{\theta} = 1\). The area of the loop is the area of a circle, which is given by: \(A = \pi r^2\) where \(r\) is the radius of the loop. The change in magnetic flux is therefore: \(\Delta \Phi = \Phi_{final} - \Phi_{initial} = B \cdot (A_{final}-A_{initial})\)

Calculate the induced EMF

Now that we have the change in magnetic flux, we can use Faraday's Law to find the induced EMF: \(\epsilon = -\frac{\Delta \Phi}{\Delta t}\) where \(\epsilon\) is the induced EMF and \(\Delta t\) is the change in time.

Calculate the average current

Finally, we can use Ohm's Law to find the average current in the loop: \(I = \frac{\epsilon}{R}\) where \(I\) is the average current and \(R\) is the resistance of the loop. We can substitute the known values to find the average current.

Step 1

Calculate the change in magnetic flux: \(\Delta \Phi = (0.426 \cdot 10^{-4} \mathrm{~T}) (\pi (25.0 \mathrm{~cm})^2 - \pi (20.0 \mathrm{~cm})^2)\).

Step 2

Calculate the induced EMF: \(\epsilon = -\frac{\Delta \Phi}{1.00 \mathrm{~s}}\).

Step 3

Calculate the average current: \(I = \frac{\epsilon}{30.0 \Omega}\). After plugging in the values from the previous steps and performing the calculations, we find the average current to be \(2.81 \times 10^{-6} A\), or \(2.81 \mathrm{~\mu A}\).