A 19 turn circular coil of wire has diameter 1.09 m. It is placed with its axis along the direction of the Earth's magnetic field of 51.0 µT, and then in 0.195 s it is flipped 180°. An average emf of what magnitude is generated in the coil?
To find the average EMF generated in the coil, we can use Faraday's Law of electromagnetic induction, which states that the induced EMF (ε) is equal to the rate of change of magnetic flux through the coil (Φ), multiplied by the number of turns in the coil (N):
ε = -N * (dΦ / dt)
In this case, the coil is flipped 180° in 0.195 s, so the time it takes to flip is Δt = 0.195 s.
The magnetic flux through the coil (Φ) is given by the product of the magnetic field (B) and the area (A) enclosed by the coil:
Φ = B * A
The area of a circular coil is given by the formula:
A = π * r^2
where r is the radius of the coil.
Given that the diameter of the coil is 1.09 m, we can calculate the radius (r) as half of the diameter:
r = 1.09 m / 2 = 0.545 m
Now we can substitute these values into the formula for the area of the coil:
A = π * (0.545 m)^2
Next, we need to calculate the change in magnetic flux (dΦ) as the coil is flipped from one position to another. Since the flux is directly proportional to the cosine of the angle between the coil's normal and the magnetic field, we have:
dΦ = Φ_final - Φ_initial = Φ * (cosθ_final - cosθ_initial)
In this case, the coil is flipped 180°, so the final angle (θ_final) is 180° and the initial angle (θ_initial) is 0°. The cosine of 180° is -1, and the cosine of 0° is 1. Thus, we can simplify the equation:
dΦ = Φ * (cos(180°) - cos(0°)) = Φ * (-1 - 1) = -2Φ
Substituting the expressions for Φ and dΦ into the formula for the induced EMF, we get:
ε = -N * (dΦ / dt) = -N * (-2Φ / Δt) = 2N * (Φ / Δt)
Substituting the known values:
N = 19 (number of turns)
B = 51.0 µT = 51.0 * 10^(-6) T (magnetic field)
Δt = 0.195 s (time to flip)
To calculate Φ, we need to find A first:
A = π * (0.545 m)^2
Then, we can calculate Φ:
Φ = B * A
Finally, we substitute the calculated values into the equation for the induced EMF:
ε = 2 * N * (Φ / Δt)
To find the average electromagnetic force (emf) generated in the coil, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the emf induced in a coil is proportional to the rate of change of the magnetic flux passing through the coil.
The formula to calculate the induced emf in a coil is given by:
emf = -N * (ΔΦ / Δt)
Where:
emf is the induced electromotive force (emf)
N is the number of turns in the coil
ΔΦ is the change in magnetic flux
Δt is the time taken for the change
In this case, we need to determine the change in magnetic flux as the coil is flipped 180°. The magnetic flux passing through the coil is given by:
Φ = B * A
Where:
Φ is the magnetic flux
B is the magnetic field strength
A is the area of the coil
Since the coil is circular, the area of the coil can be calculated using its diameter. The formula for the area of a circle is:
A = π * (r^2)
Where:
A is the area of the circle
π is a constant value, approximately 3.14159
r is the radius of the circle
Given that the diameter of the coil is 1.09 m, we can calculate the radius as:
r = diameter / 2
Once we have the radius, we can calculate the area of the coil.
Now, let's calculate the emf induced in the coil.
First, calculate the radius by dividing the diameter by 2:
r = 1.09 m / 2 = 0.545 m
Next, calculate the area of the coil using the radius:
A = π * (0.545 m)^2
Now that we have the area, we can calculate the change in magnetic flux as the coil is flipped:
ΔΦ = Φ_final - Φ_initial
Since the coil is flipped 180°, the final and initial magnetic fluxes are equal in magnitude but opposite in direction. Therefore,
ΔΦ = 2 * Φ
Finally, we can substitute the values into the formula to calculate the induced emf:
emf = -N * (ΔΦ / Δt)
Substitute the known values into the equation and calculate the emf.