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.
The formula for the induced emf in a circular loop is given by:
emf = (μ₀ * N * A * dI) / (2 * R)
Where:
- emf is the induced emf
- μ₀ is the permeability of free space (4π × 10^(-7) T⋅m/A)
- N is the number of turns in the loop
- A is the area of the loop
- dI is the change in current
- R is the radius of the loop
Given:
- emf = 983 nano-volts = 983 × 10^(-9) volts
- dI = 200 amps/second
- R₁ = 1 meter
- R₂ = 5 centimeters = 0.05 meters
Let's calculate the separation of the loops step-by-step:
1. Convert the emf to volts:
emf = 983 × 10^(-9) volts
2. Substitute the values into the formula:
983 × 10^(-9) volts = (4π × 10^(-7) T⋅m/A) * (1 turn) * (π * (0.05 meters)²) * (200 amps/second) / (2 * 1 meter)
3. Simplify the expression:
983 × 10^(-9) volts = (4π × 10^(-7) T⋅m/A) * (1 turn) * (π * 0.0025 meters²) * (200 amps/second) / (2 * 1 meter)
4. Rearrange the formula to solve for the separation of the loops (d):
d = (2 * emf * R₁) / ((μ₀ * N * A * dI) - (4π * R₂ * emf))
5. Substitute the known values into the formula:
d = (2 * (983 × 10^(-9) volts) * (1 meter)) / ((4π × 10^(-7) T⋅m/A) * (1 turn) * (π * 0.0025 meters²) * (200 amps/second) - (4π * 0.05 meters * (983 × 10^(-9) volts)))
6. Calculate the separation of the loops (d):
d = (1.966 × 10^(-6) volts * meter) / ((4π × 10^(-7) T⋅m/A) * (1 turn) * (π * 0.0025 meters²) * (200 amps/second) - (4π * 0.05 meters * (983 × 10^(-9) volts)))
7. Calculate the final answer:
d ≈ 1.63 × 10^(-2) meters
Therefore, the separation of the loops is approximately 0.0163 meters or 16.3 millimeters.
To find the separation between the two loops, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced emf in a loop is given by the equation:
Emf = -N * d(Flux) / dt
Where:
Emf is the induced emf (in volts)
N is the number of turns in the loop
d(Flux) / dt is the rate of change of magnetic flux through the loop
In this case, the induced emf is given as 983 nano-volts (983 x 10^-9 V) and the rate of change of magnetic flux is the quantity we need to determine.
The magnetic flux through a loop is given by the equation:
Flux = B * A
Where:
B is the magnetic field (in teslas)
A is the area of the loop (in square meters)
Since the loops are coaxial, the area of the small loop is equal to the area of the larger loop. The area of a loop is defined as A = π * r^2, where r is the radius.
Given:
Radius of the larger loop (R) = 1 meter
Radius of the smaller loop (r) = 0.05 meters (5 centimeters)
Area of the large loop (A) = π * R^2
Area of the small loop = π * r^2
To find the magnetic field, we need to calculate the rate of change of current through the large loop (dl/dt). We are given that the current is increasing linearly at a rate of 200 amps/second.
Now, let's calculate the values step by step:
Step 1: Determine the area of the large loop
A = π * R^2
A = π * (1 meter)^2
A = π square meters
Step 2: Calculate the rate of change of current through the large loop
dl/dt = 200 amps/second
Step 3: Calculate the magnetic field (B) through the large loop
From Faraday's law, we can rearrange the equation as:
B = -Emf / (N * dA / dt)
Substituting the given values:
B = -(983 x 10^-9 V) / (1 * dA / dt)
Step 4: Determine the rate of change of magnetic field (dB/dt)
Since the current is increasing with time, the rate of change of magnetic field (dB/dt) is equal to the rate of change of current (dl/dt).
dB/dt = 200 amps/second
Step 5: Calculate the rate of change of magnetic flux (d(Flux) / dt) through the loop
d(Flux) / dt = dB/dt * A
Step 6: Determine the separation between the loops
Since the loops are coaxial, the separation between them is equal to the radius of the larger loop minus the radius of the smaller loop.
Separation = R - r
Now, let's calculate the values:
Step 1: Determining the area of the large loop
A = π * (1 meter)^2
A = π square meters (approximately 3.1416 square meters)
Step 2: Calculating the rate of change of current through the large loop
dl/dt = 200 amps/second
Step 3: Calculating the magnetic field (B) through the large loop
B = -(983 x 10^-9 V) / (1 * dA / dt)
Step 4: Determining the rate of change of magnetic field (dB/dt)
dB/dt = 200 amps/second
Step 5: Calculating the rate of change of magnetic flux (d(Flux) / dt) through the loop
d(Flux) / dt = dB/dt * A
Step 6: Determining the separation between the loops
Separation = R - r
Note: The calculation involves several steps and intermediate values to maintain accuracy and significant figures.