To monitor the breathing of a hospital patient, a thin belt is girded around the patient's chest . The belt is a 215-turn coil. When the patient inhales, the area encircled by the coil increases by 41.0 cm2. The magnitude of Earth's magnetic field is 50.0 µT and makes an angle of 30.0° with the plane of the coil. Assuming a patient takes 2.10 s to inhale, find the magnitude of the average induced emf in the coil during that time.
|| = µV
Flux change/Time = N*B*A*sin30/2.1s
= N*(50*10^-6T)*(41*10^-4m^2)*(0.5)/2.1s
= (215)*1.95*10^-8 V
= 4.2 µV
To find the magnitude of the average induced emf in the coil, we can use Faraday's Law of electromagnetic induction, which states that the induced emf in a circuit is equal to the rate of change of magnetic flux through the circuit.
The formula for the induced emf is given by:
ℰ = -N * dΦ/dt
Where:
ℰ is the induced emf
N is the number of turns of the coil
dΦ/dt is the rate of change of magnetic flux through the coil
First, we need to find the rate of change of magnetic flux through the coil. The magnetic flux is given by:
Φ = B * A * cos(θ)
Where:
B is the magnitude of the Earth's magnetic field
A is the area encircled by the coil
θ is the angle between the magnetic field and the plane of the coil
Given:
B = 50.0 µT
A = 41.0 cm² = 0.0041 m²
θ = 30.0°
We can substitute these values into the equation to find the initial magnetic flux, Φ_initial:
Φ_initial = B * A * cos(θ)
Now, we need to find the final magnetic flux when the area increases by 41.0 cm². The final area, A_final, is given by:
A_final = A + 41.0 cm² = A + 0.0041 m²
We can substitute the final area into the equation to find the final magnetic flux, Φ_final:
Φ_final = B * A_final * cos(θ)
The change in magnetic flux, ΔΦ, is given by:
ΔΦ = Φ_final - Φ_initial
Finally, we can substitute the values into the formula for induced emf to find the magnitude of the average induced emf during the inhalation period. Remember to convert the units to the appropriate values (microvolts - µV):
ℰ = -N * (ΔΦ / Δt) = -N * (ΔΦ / 2.10 s)
So, with these calculations, you can find the magnitude of the average induced emf in the coil during the patient's inhalation period.