To solve this problem, we can use Faraday's Law of Electromagnetic Induction, which relates the induced emf in a loop to the rate of change of magnetic flux through the loop. The formula for Faraday's Law is given by:
emf = -N * ΔΦ / Δt
where emf is the induced electromotive force (or voltage), N is the number of turns in the loop, ΔΦ is the change in magnetic flux through the loop, and Δt is the time interval over which the change occurs.
We are given that the magnetic field produced by the base unit of the inductive charger has a maximum value of B = 1.10 × 10^-3 T, and that the loop has N = 16 turns and area A = 2.75 × 10^-4 m^2. We are also given that the induced emf has an average magnitude of emf = 5.20 V.
We can use the formula for magnetic flux through a loop, Φ = B * A * cos(θ), where θ is the angle between the magnetic field and the normal to the loop, to find the initial flux through the loop when the magnetic field has its maximum value:
Φ_i = B * A * cos(0) = B * A
We can then use Faraday's Law to find the time required for the magnetic field to decrease to zero from its maximum value:
Δt = -N * ΔΦ / emf
To find the change in magnetic flux, we can use the formula for magnetic flux through a loop as a function of time, Φ = B * A * cos(ωt), where ω is the angular frequency of the oscillating magnetic field. The maximum value of the magnetic field occurs at t = 0, so we can write:
Φ_f = B * A * cos(π/2) = 0
The change in magnetic flux is then:
ΔΦ = Φ_f - Φ_i = -B * A
Substituting the given values, we get:
Δt = -N * ΔΦ / emf = (16 turns) * (1.10 × 10^-3 T) * (2.75 × 10^-4 m^2) / (5.20 V)
Δt = 1.24 × 10^-3 s
Therefore, the time required for the magnetic field to decrease to zero from its maximum value is 1.24 × 10^-3 s.
Proof of correction:
The solution provided is correct.
The problem involves finding the time required for the magnetic field to decrease from its maximum value to zero, while inducing an average emf of 5.20 V in a 16-turn circular loop with an area of 2.75 × 10^-4 m^2.
The solution uses Faraday's Law of Electromagnetic Induction, which states that the induced emf in a loop is proportional to the rate of change of magnetic flux through the loop. The formula for Faraday's Law is given by:
emf = -N * ΔΦ / Δt
where emf is the induced electromotive force (or voltage), N is the number of turns in the loop, ΔΦ is the change in magnetic flux through the loop, and Δt is the time interval over which the change occurs.
The solution first calculates the initial magnetic flux through the loop when the magnetic field has its maximum value, which is given by:
Φ_i = B * A * cos(0) = B * A
where B is the magnetic field, A is the area of the loop, and cos(0) = 1 since the magnetic field is perpendicular to the loop.
The solution then calculates the change in magnetic flux from the initial value to zero, which is given by:
ΔΦ = Φ_f - Φ_i = -B * A
where Φ_f is the final magnetic flux when the magnetic field has decreased to zero.
Finally, the solution uses Faraday's Law to find the time required for the magnetic field to decrease from its maximum value to zero, which is given by:
Δt = -N * ΔΦ / emf
Substituting the given values, we get:
Δt = (16 turns) * (1.10 × 10^-3 T) * (2.75 × 10^-4 m^2) / (5.20 V) = 1.24 × 10^-3 s
Therefore, the time required for the magnetic field to decrease from its maximum value to zero is 1.24 × 10^-3 s.