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.
A metal wire loop hangs near the earth's surface from an insulating string. The loop spins continuously (at constant angular velocity) so that its area vector points north, then west, then south, then east, etc.
4.A conducting slider (thick line) moves without friction along a vertical U-shaped conducting track (thin line) in a magnetic field as shown.
To find the separation between the two loops, we can use Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in a loop is equal to the rate of change of magnetic flux through the loop.
The magnetic flux through a loop is given by the equation:
Φ = B * A * cos(θ)
Where:
Φ is the magnetic flux
B is the magnetic field
A is the area of the loop
θ is the angle between the magnetic field and the normal to the loop
In this case, the large loop is driven with a linearly increasing current, creating a changing magnetic field. The induced emf in the small loop is given as 983 nanovolts (983 x 10^-9 V). We need to find the separation between the loops.
First, we need to find the magnetic field at the location of the small loop due to the current in the large loop.
The magnetic field at the center of the large loop can be found using Ampere's Law:
B = (μ₀ * I) / (2 * π * r)
Where:
B is the magnetic field
μ₀ is the magnetic constant (4π × 10^-7 Tm/A)
I is the current in the large loop
r is the radius of the large loop
Substituting the given values:
B = (4π × 10^-7 Tm/A) * (200 A/s) / (2π * 1 m)
B = 2 × 10^-7 T
Next, we need to calculate the magnetic flux through the small loop. Since the two loops are coaxial, the area of the small loop is the same as the large loop. The angle θ between the magnetic field and the normal to the loop is 0 degrees since they are aligned.
Φ = B * A * cos(0°)
Φ = B * A
To find the separation between the loops, we can rearrange the equation Φ = B * A to solve for the distance (d):
d = Φ / (B * A)
Substituting the given values:
d = (983 × 10^-9 V) / ((2 × 10^-7 T) * π * (0.05 m)^2)
d ≈ 3.15 m
Therefore, the separation between the loops is approximately 3.15 meters.