A wire is bent into the shape of an planar Archimedean spiral which in polar coordinates is described by the equation
r=bθ.
The spiral has N=100 turns and outer radius
R=10 cm (R is the distance from point O to point T). Note that in the figure below we show a spiral having only 3 turns. The circuit is placed in a homogeneous magnetic field perpendicular to the plane of the spiral. The time dependence of the magnetic field induction is given by
B=B_0 cos (ωt)
where
B_0=1 μT and ω=2×10^6 s−1
Determine, the amplitude of the emf in Volts induced in the circuit.
ℰ = -N•dΦ/dt = -N•d(B₀Scosωt)/dt =
= NB₀Sωsinωt,
ℰ max= NB₀Sω,
Without the figure, I believe that S=πR² =>
ℰ max= NB₀πR²ω =
=100•10⁻⁶•3.14•0.1²•2•10⁶ =6.28 V
To determine the amplitude of the induced emf in the circuit, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced emf in a circuit is equal to the rate of change of magnetic flux through the circuit.
The magnetic flux through the circuit can be calculated by integrating the product of the magnetic field induction (B) and the area (A) enclosed by the circuit. Since the spiral is planar and the magnetic field is perpendicular to the plane of the spiral, the area enclosed by each turn of the spiral is constant.
The area enclosed by each turn of the spiral can be calculated by subtracting the area of the inner circle from the area of the outer circle. Let's calculate the area of each turn and the total magnetic flux through the entire spiral.
The outer radius of the spiral is given as R = 10 cm. Since there are N = 100 turns, the total length of the wire (L) can be calculated by multiplying the circumference of each turn by the number of turns.
Circumference of each turn = 2πR
Length of wire = L = N * 2πR
The rate of change of magnetic flux (dΦ/dt) can be obtained by differentiating the magnetic flux equation with respect to time (t).
dΦ/dt = d(BA)/dt = (dB/dt) * A
Substituting the given values:
B_0 = 1 μT = 1 * 10^(-6) T
ω = 2 * 10^6 s^(-1)
The induced emf (ε) is then given by:
ε = -dΦ/dt
Now, to calculate the amplitude of the induced emf, we disregard the negative sign and substitute the appropriate values into the equation.
ε = |(dB/dt) * A|
First, let's calculate the area (A) enclosed by each turn of the spiral:
A = π * (R^2 - (R - b)^2)
where b is the distance between adjacent turns of the spiral. Since the spiral is described by the equation r = bθ, we can find b by considering the outer radius (R) and the number of turns (N).
b = R/N
Substituting the value of b, we can calculate the area:
A = π * (R^2 - (R - R/N)^2)
Now that we have the area, let's calculate the amplitude of the induced emf by substituting the values of (dB/dt) and A into the equation:
ε = |(B_0 * ω) * A|
Substituting the given values:
B_0 = 1 μT = 1 * 10^(-6) T
ω = 2 * 10^6 s^(-1)
R = 10 cm = 0.1 m
N = 100
Now you can substitute these values into the formulas and calculate the amplitude of the induced emf in volts.