A uniform magnetic field makes an angle of 30 degrees with the axis of a circular loop of 300 turns and radius of 4 cm. The magnetic field changes at a rate of 85 T/s. Determine the magnitude of the induced emf in the loop.
To determine the magnitude of the induced emf in the loop, we can use Faraday's law of electromagnetic induction, which states that the induced emf (ε) is equal to the rate of change of magnetic flux through the loop.
The magnetic flux (Φ) through a loop is given by the product of the magnetic field (B), the area of the loop (A), and the cosine of the angle between the magnetic field and the normal to the loop. Mathematically, it is expressed as:
Φ = B * A * cos(θ)
Where:
B = Magnetic field
A = Area of the loop
θ = Angle between the magnetic field and the normal to the loop
In this case, we are given:
B = 85 T/s (Rate of change of magnetic field)
A = π * r^2 (Area of the circular loop, where r = 4 cm = 0.04 m)
θ = 30° (Angle between magnetic field and the normal to the loop)
Substituting these values into the equation, we have:
Φ = (85 T/s) * (π * (0.04 m)^2) * cos(30°)
Now, we can calculate the magnetic flux:
Φ = (85 T/s) * (π * 0.0016 m^2) * cos(30°)
= (85 T/s) * (π * 0.0016 m^2) * √3/2 (since cos(30°) = √3/2)
≈ 0.272 T⋅m^2/s
Finally, we can determine the magnitude of the induced emf using Faraday's law:
ε = -dΦ/dt
Since the magnetic flux is changing at a rate of 0.272 T⋅m^2/s, the induced emf is equal to:
ε = -dΦ/dt
= -0.272 T⋅m^2/s
Therefore, the magnitude of the induced emf in the loop is approximately 0.272 volts.