An increasing magnetic field is 45.0° clockwise from the vertical axis, and increases from 0.500 T to 0.6 T in 4.30 s. There is a coil at rest whose axis is along the vertical and it has 100 turns and a diameter of 6.00 cm. What is the induced emf?

To find the induced emf, we can use Faraday's law of electromagnetic induction, which states that the induced emf in a closed loop equals the negative of the rate of change of the magnetic flux through the loop.

The formula for Faraday's law is:

emf = -N * ΔΦ/Δt

where: N = number of turns in the coil, ΔΦ = change in magnetic flux, Δt = change in time.

First, we need to find the change in magnetic flux (ΔΦ). The magnetic flux (Φ) through a loop is given by the dot product of the magnetic field (B) and the area vector (A), which is:

Φ = B * A * cos(θ)

where: B = magnetic field, A = area of the loop, θ = angle between B and A.

Given that the magnetic field increases from 0.500 T to 0.6 T, the change in magnetic field (ΔB) is 0.6 T - 0.500 T = 0.1 T.

The area of the coil (A) can be calculated using the formula for the area of a circle (πr²), where r is the radius of the coil. Given that the diameter of the coil is 6.00 cm, the radius is 6.00 cm / 2 = 3.00 cm = 0.03 m. So, A = π * (0.03 m)² = 0.002827 m².

The angle between the magnetic field and the area vector (θ) is 45.0°. However, since the magnetic field is increasing, the angle is actually 45.0° - 90° = -45.0°.

So, the change in magnetic flux (ΔΦ) is:

ΔΦ = ΔB * A * cos(θ) = 0.1 T * 0.002827 m² * cos(-45.0°) = 0.0002 Wb (weber).

The change in time (Δt) is given as 4.30 s.

Finally, we can substitute these values into Faraday's law to find the induced emf:

emf = -N * ΔΦ/Δt = -100 turns * 0.0002 Wb / 4.30 s = -0.00465 V or -4.65 mV.

The negative sign indicates that the induced emf opposes the change in the original magnetic field, as per Lenz's law.