Would someone check this please?
A circular loop of wire of radius 0.10 m and of resistance 2.0 �~ 10−3 ƒ¶ is in
a region where there is a uniform magnetic field B.
The field is at 30�‹ to the normal to the plane of the wire loop as shown in Figure 3. The magnitude of the
magnetic field falls at a steady rate from 6.0 �~ 10−2 T to zero in 2.0 seconds.
What is the value of the current flowing round the wire loop while the field is decreasing?
I get 0.8A
To find the value of the current flowing around the wire loop while the magnetic field is decreasing, you can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced electromotive force (EMF) in a closed loop is equal to the rate of change of magnetic flux through the loop.
The equation for the induced EMF is given by:
EMF = -N * (dΦ/dt), where N is the number of turns in the wire loop, dΦ/dt is the rate of change of magnetic flux through the loop.
In this case, the magnetic field is decreasing at a steady rate from 6.0 × 10^(-2) T to zero in 2.0 seconds. The change in magnetic field, ΔB, is given by:
ΔB = 6.0 × 10^(-2) T - 0 T = -6.0 × 10^(-2) T
The time interval, Δt, is given as 2.0 seconds.
Now, you need to calculate the rate of change of magnetic flux, dΦ/dt. The magnetic flux through a circular loop can be calculated using the equation:
Φ = B * A * cos(θ), where B is the magnetic field, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the loop.
In this case, the radius of the wire loop is given as 0.10 m. Therefore, the area of the loop, A, can be calculated as:
A = π * r^2 = π * (0.10 m)^2
The angle θ is given as 30°, but we need to convert it to radians for the equation. Radians can be calculated by converting degrees using the formula:
θ (Radians) = θ (Degrees) * π / 180
Substituting the values, we get:
θ (Radians) = 30° * π / 180 = π / 6 radians
Now, we can calculate the magnetic flux at the initial field strength:
Φ_initial = B_initial * A * cos(θ)
Φ_initial = (6.0 × 10^(-2) T) * (π * (0.10 m)^2) * cos(π / 6)
Similarly, we can calculate the magnetic flux at the final field strength (zero):
Φ_final = B_final * A * cos(θ)
Φ_final = (0 T) * (π * (0.10 m)^2) * cos(π / 6)
The rate of change of magnetic flux, dΦ/dt, is then given by:
dΦ/dt = (Φ_final - Φ_initial) / Δt
Once you have calculated dΦ/dt, you can use Faraday's law to find the induced EMF. Since the wire loop has a resistance of 2.0 × 10^(-3) Ω, you can use Ohm's law, V = I * R, to find the current flowing through the loop while the magnetic field is decreasing.
Now, plug in the values and calculate the current, I, using the equation:
EMF = -N * (dΦ/dt)
V = I * R
Finally, rearrange the equation to solve for I:
I = V / R
The value of the current flowing round the wire loop while the field is decreasing should be the calculated value of I.