A circular loop of wire with radius 0.100 m and resistance R = 5.00 Ω. The loop is placed in a magnetic field of strength B = 0.125 T, which can be rotated. The angle of the magnetic field with respect to the normal of the loop is θ(t), where t denotes the time. Initially, θ(0) = 2/π rad . The angular speed at which the magnetic field rotates is w= 60.0rads−1.
(a) Find an expression for the magnetic flux through the loop at time t.
(b) Find an expression for the magnitude of the induced EMF in the loop at time t.
b) Induced EMF = -(Rate of change of Flux / time)
Differentiate the flux function with respect to t, to obtain the expression.
Note that the negative sign indicates that the EMF is induced in the opposite direction, as per Lenz's Law.
Let us know if you do not know what "differentiate" means. There is another way.
Thank you both, I understand differentiation however I would be interested to see the other way for completeness should you have time.
My only question is how would I best relate the R for resistance to the magnetic flux equation would I be using faradays law here?
To find the expression for the magnetic flux through the loop at time t, we'll first need to calculate the magnetic field passing through the loop. The magnetic field passing through the loop is given by B = B₀ cos(θ), where B₀ is the maximum value of the magnetic field.
Given that B = 0.125 T and θ(t) = ωt + θ₀, where ω is the angular speed and θ₀ is the initial angle, we can find the expression for B(t) as B(t) = B₀ cos(ωt + θ₀).
Next, we'll calculate the area of the loop to find the expression for the magnetic flux. The area of a circular loop is given by A = πr², where r is the radius. Given r = 0.100 m, the area of the loop is A = π(0.100 m)² = 0.0314 m².
The magnetic flux (Φ) passing through the loop at time t is given by the product of the magnetic field and the area: Φ(t) = B(t) * A.
Substituting the expressions for B(t) and A, we get:
Φ(t) = (B₀ cos(ωt + θ₀)) * 0.0314 m².
Therefore, the expression for the magnetic flux through the loop at time t is Φ(t) = 0.0314 B₀ cos(ωt + θ₀) m².
To find the expression for the magnitude of the induced electromotive force (EMF) in the loop at time t, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced EMF (ε) is given by the rate of change of magnetic flux with respect to time: ε(t) = -dΦ(t)/dt.
Differentiating Φ(t) with respect to time, we get:
dΦ(t)/dt = -0.0314 B₀ ω sin(ωt + θ₀).
Therefore, the expression for the magnitude of the induced EMF in the loop at time t is ε(t) = 0.0314 B₀ ω sin(ωt + θ₀) V.
a) Magnetic Flux = Dot product of Magnetic Field Vector and Area Vector
=> Flux = B * A
= BAcosθ
At time t, θ = θ(0) + wt
= (2/π) + 60t
Plug this value into the equation for flux.