A 39-turn circular coil of wire has a diameter of 1.2m. It is placed with its axis along the direction of the Earth's magnetic field of 58μT at 52° below the horizontal, and then in 0.16s it is flipped 180°.
a) What is volts the magnitude of the average induced emf in the coil?
b) We would like 100 x the voltage found in part a to be generated between the wingtips of an airplane of 62.9 m wingspan flying horizontally in the same region. What would be in m/s the needed speed of the airplane?
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a) To find the magnitude of the average induced emf in the coil, we can use Faraday's law of electromagnetic induction.
The equation for the induced emf is given by:
emf = -N(dΦ/dt)
Where:
emf = induced electromotive force
N = number of turns in the coil
dΦ/dt = rate of change of magnetic flux
The magnetic flux (Φ) through a circular coil with a diameter of 1.2m can be calculated as:
Φ = B * A
Where:
B = magnetic field strength
A = area of the coil
The area of a circle is given by:
A = π * (r^2)
Given that the diameter (d) of the coil is 1.2m, the radius (r) is half of that:
r = d/2
So,
r = 1.2/2 = 0.6m
Plugging in the values, the area of the coil is:
A = π * (0.6^2)
Next, we need to calculate the rate of change of magnetic flux (dΦ/dt). Since the coil is flipped 180° in 0.16s, the time it takes to change the magnetic flux is Δt = 0.16s.
Finally, we can calculate the emf using the equation mentioned earlier.
emf = -N * (dΦ/dt)
Substituting the values, we can find the magnitude of the average induced emf in the coil.
b) To find the needed velocity of the airplane, we need to use the equation relating the induced emf and the velocity.
The induced emf (emf) in the airplane is given by:
emf_airplane = B * L * v
Where:
B = magnetic field strength
L = wingspan of the airplane
v = velocity of the airplane
We want the emf_airplane to be 100 times the emf calculated in part a.
By substituting the values, we can solve for the velocity.
Note: The direction of the induced emf might be opposite for a circular coil depending on the direction of the magnetic field. In this case, make sure to consider the signs while calculating the emf.
To find the magnitude of the average induced electromotive force (emf) in the coil, we can use Faraday's law of electromagnetic induction:
emf = -N * dΦ/dt
where emf is the induced electromotive force, N is the number of turns in the coil, and dΦ/dt is the rate of change of magnetic flux. To calculate the induced emf, we need to find the rate of change of magnetic flux.
First, we need to determine the initial and final magnetic flux through the coil.
The initial magnetic flux through the coil is given by:
Φ_initial = B_initial * A
where B_initial is the initial magnetic field strength and A is the area of the coil.
The final magnetic flux through the coil is given by:
Φ_final = B_final * A
where B_final is the final magnetic field strength.
Since the coil is flipped 180°, the final and initial magnetic fields are equal in magnitude but opposite in direction. Therefore, B_final = -B_initial.
The area of the coil, A, can be calculated using the formula for the area of a circle:
A = π * (radius)^2
In this case, the diameter is given, so we need to calculate the radius first:
radius = diameter / 2
Once we have the radius, we can calculate the area.
Now we can substitute the values into the formulas to find the initial and final magnetic flux:
Φ_initial = B_initial * π * (radius)^2
Φ_final = -B_initial * π * (radius)^2
Next, we need to calculate the rate of change of magnetic flux (dΦ/dt). Since the coil is flipped in 0.16s, the change in flux occurs over that time period. Hence, the rate of change of magnetic flux is:
dΦ/dt = (Φ_final - Φ_initial) / Δt
where Δt is the change in time.
Once we have the rate of change of magnetic flux, we can calculate the average induced emf using the formula:
emf = -N * dΦ/dt
a) To find the magnitude of the average induced emf in the coil, substitute the known values into the equation and calculate the value.
b) To find the needed speed of the airplane, we need to use the concept of electromagnetic induction. The magnetic field strength B and the wingspan of the airplane are given. We know that the induced emf is directly proportional to the product of the magnetic field strength, the wingspan, and the velocity of the airplane:
emf = k * B * wingspan * velocity
where k is a constant.
We can rearrange the equation to solve for the velocity:
velocity = emf / (k * B * wingspan)
Substitute the given values of emf, wingspan, B, and calculate the velocity required.