A 500 turn solenoid with a length of 20 cm and a radius of 1.5 cm carries a current of 2.0 A. A second coil of four turns is wrapped tightly about this solenoid so that it can be considered to have the same radius as the solenoid. Find the following when the current in the solenoid increases to 5.0 A in a period of 0.90 s.
(a) the change in the magnetic flux through the coil
T·m2
(b) the magnitude of the average induced emf in the coil
V
I need to see your thinking. These are very basic questions about computing magnetic flux from current in a solenoid, and from the changing B calculating the emf generated using faradays law. I need to see what you are missing.
To find the change in the magnetic flux through the coil, we need to know the initial and final values of the magnetic field.
The magnetic field inside a solenoid can be given by the formula:
B = μ₀ * n * I
where B is the magnetic field, μ₀ is the permeability of free space (4π x 10^-7 T·m/A), n is the number of turns per unit length, and I is the current.
Given that the solenoid has 500 turns, a length of 20 cm (0.2 m), and carries a current of 2.0 A, we can calculate the initial magnetic field inside the solenoid using the formula:
B_initial = μ₀ * n_initial * I_initial
To find n_initial, we can divide the total number of turns (500) by the length of the solenoid (0.2 m):
n_initial = 500 / 0.2 = 2500 turns/m
Substituting the values into the formula:
B_initial = (4π x 10^-7 T·m/A) * (2500 turns/m) * (2.0 A)
Now, to find the final magnetic field, we can use the same formula with the new current value (5.0 A).
B_final = μ₀ * n_final * I_final
Since the number of turns and radius remain the same, n_final and B_final will be the same as n_initial and B_initial (assuming the second coil does not affect the magnetic field significantly).
Now we can calculate the change in magnetic flux (ΔΦ) using the formula:
ΔΦ = B_final * A - B_initial * A
where A is the cross-sectional area of the solenoid (π * r^2).
Given that the radius of the solenoid is 1.5 cm (0.015 m), we can calculate the change in magnetic flux:
ΔΦ = B_final * A - B_initial * A
= (B_final - B_initial) * A
= (B_final - B_initial) * (π * r^2)
To find the magnitude of the average induced emf in the coil, we can use Faraday's law of electromagnetic induction:
ε = -(ΔΦ / Δt)
where ε is the induced emf, ΔΦ is the change in magnetic flux, and Δt is the change in time.
Given that the current increases to 5.0 A in a period of 0.90 s, we can substitute the values into the equation to find the magnitude of the average induced emf (ε).