A very large loop of metal wire with radius 1 meter is driven with a linearly increasing current at a rate of 200 amps/second. A very small metal wire loop with radius 5 centimeter is positioned a small distance away with its center on the same axis (the loops are coaxial). The small loop experiences an induced emf of 983 nano-volts. What is the separation of the loops in meters? Note that a subtraction step in the solution makes it sensitive to significant figures. Keep at least four figures in your calculation.
To determine the separation of the loops, we can use Faraday's Law of electromagnetic induction. This law states that the induced electromotive force (emf) in a closed loop is equal to the negative rate of change of magnetic flux through the loop.
The magnetic flux through a loop is given by the formula:
Φ = B * A * cos(θ)
Where:
Φ = magnetic flux
B = magnetic field strength
A = area of the loop
θ = angle between the magnetic field and the normal to the loop
In this case, we can assume that the magnetic field is uniform around the larger loop. The magnetic field at the center of the smaller loop can be calculated using the Biot-Savart Law, which relates the magnetic field produced by a current-carrying wire to the current through the wire, the distance from the wire, and the geometry of the wire.
Given that the radius of the larger loop is 1 meter and the radius of the smaller loop is 5 centimeters (or 0.05 meters), we can calculate the area of the smaller loop as follows:
A_small = π * (0.05)^2 = 0.00785 square meters
The rate of change of current through the larger loop is given as 200 amps/second. We can use this to calculate the rate of change of magnetic field at the center of the smaller loop, which is the same as the rate of change of the magnetic flux:
dB_dt = μ₀ * N * (dI_dt) / (2 * R)
Where:
dB_dt = rate of change of magnetic field
μ₀ = permeability of free space
N = number of turns in the larger loop
dI_dt = rate of change of current through the larger loop
R = distance between the loops (unknown)
The permeability of free space (μ₀) is a constant value equal to 4π × 10^(-7) T·m/A.
Let's calculate dB_dt:
dB_dt = (4π × 10^(-7) T·m/A) * 1 * (200 A/s) / (2 * R)
= (8π × 10^(-7)) / R T·m/A²
As mentioned, the induced emf in the smaller loop is given as 983 nano-volts (or 983 × 10^(-9) volts). This is equal to the rate of change of the magnetic flux:
dΦ_dt = dB_dt * A_small
Substituting the values, we have:
(983 × 10^(-9) V) = (8π × 10^(-7)) / R T·m/A² * 0.00785 m²
Now we can solve for the separation (R):
R = (8π × 10^(-7)) / [(983 × 10^(-9) V) / (0.00785 m²)]
= (8 * 3.14 * 10^(-7)) / [(983 × 10^(-9)) / (0.00785)]
≈ 0.00243 meters
Therefore, the separation between the loops is approximately 0.00243 meters.