A small circular washer of radius 0.5cm is held directly below a long straight wire carrying currrent 10A. The washer is located 0.5m above the top of table. If the washer is dropped from rest, what is the magnitude of the average induced emf in the washer from the time it is released to the moment it hits the tabletop?Assume that the magnetic field is nearly constant over the area of the washer, and equal to the magnetic field at the center of the washer.

To find the average induced electromotive force (emf) in the washer as it falls, we can use Faraday's law of electromagnetic induction. This law states that the induced emf in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Let's break down the problem step by step.

Understanding the Magnetic Field

First, we need to determine the magnetic field created by the long straight wire carrying a current. The magnetic field (B) around a long straight wire is given by the formula:

• B = (μ₀ * I) / (2 * π * r)

Where:

• μ₀ is the permeability of free space (approximately 4π × 10^-7 T·m/A),
• I is the current (10 A in this case),
• r is the distance from the wire to the center of the washer.

Calculating the Distance

Since the washer is 0.5 m above the table and we assume the wire is also at the table level, the distance (r) from the wire to the washer is simply 0.5 m.

Finding the Magnetic Field

Now, substituting the values into the formula:

• B = (4π × 10^-7 T·m/A * 10 A) / (2 * π * 0.5 m)

After simplifying, we find:

• B = (4 × 10^-7 T·m/A * 10) / (1)
• B = 4 × 10^-6 T or 4 µT.

Calculating the Change in Magnetic Flux

The magnetic flux (Φ) through the washer is given by the formula:

• Φ = B * A

Where A is the area of the washer. The area (A) of a circular washer can be calculated using:

• A = π * r²

For our washer with a radius of 0.5 cm (0.005 m):

• A = π * (0.005 m)² = π * 25 × 10^-6 m² ≈ 7.85 × 10^-5 m².

Calculating Initial Magnetic Flux

Now, we can find the initial magnetic flux (Φ_initial) when the washer is released:

• Φ_initial = B * A = 4 × 10^-6 T * 7.85 × 10^-5 m² ≈ 3.14 × 10^-10 Wb.

Final Magnetic Flux

As the washer falls, the distance from the wire decreases. When it reaches the tabletop, the distance (r) becomes 0 m, leading to a magnetic field of:

• B_final = (μ₀ * I) / (2 * π * 0) = ∞ (undefined, but we consider it very large).

However, for practical purposes, we can consider the change in flux over the distance it falls. The average magnetic flux change can be approximated by considering the average distance from the wire during the fall.

Calculating Average Induced EMF

The average induced emf (ε) can be calculated using the formula:

• ε = -ΔΦ/Δt

Where ΔΦ is the change in magnetic flux and Δt is the time taken to fall. The time (t) taken to fall from 0.5 m can be calculated using the equation of motion:

• t = √(2h/g)

Where h is the height (0.5 m) and g is the acceleration due to gravity (approximately 9.81 m/s²):

• t = √(2 * 0.5 m / 9.81 m/s²) ≈ 0.32 s.

Final Calculation of Induced EMF

Now, we can find the change in magnetic flux:

• ΔΦ = Φ_final - Φ_initial ≈ 0 - 3.14 × 10^-10 Wb = -3.14 × 10^-10 Wb.

Substituting into the emf formula:

• ε = -(-3.14 × 10^-10 Wb) / 0.32 s ≈ 9.81 × 10^-10 V.

Thus, the magnitude of the average induced emf in the washer from the time it is released until it hits the tabletop is approximately 9.81 × 10^-10 volts.