An infinitely-long wire with resistivity 2.0*10^-8 ohm*m initially has 1.5 A of "conduction" current flowing through it. When the emf connected to the wire is turned off, this current stops in time of .6ns. Estimate the amount of displacement current created in the wire as the conduction current stops. Assume the dielectric constant of the wire is equal to one. Also, how would the magnitude of the magnetic field created by the displacement current compare to the field created by the conduction? Explain.
So for the first question I did 1.5A=1*(8.85*106-12 C^2/(N*m^2))*(BAcos(theta)/(6*10^-10s) and came up with 101.69=BAcos(theta), but I know this isn't right. What did I do wrong? For the second question I think you use the formula E0*(dE/dt), how di I know what dE is?
To answer the first question, you started with the correct equation from the Maxwell's equations, relating conduction current with displacement current. However, there seems to be a mistake in your calculation.
The equation you used is derived from Ampere's law, which states that the line integral of the magnetic field around a closed loop is equal to the sum of the conduction current passing through that loop and the displacement current through that loop. Mathematically, it can be written as:
∮ B · dl = μ₀(I_conduction + I_displacement),
where B is the magnetic field, dl is an infinitesimal element of length on the loop, μ₀ is the vacuum permeability (4π × 10^-7 T·m/A), I_conduction is the conduction current, and I_displacement is the displacement current.
For an infinitely long wire, the magnetic field is circular and has a constant magnitude everywhere on the loop. Thus, you can take the magnitude of the magnetic field outside the integral, resulting in:
B ∮ dl = μ₀(I_conduction + I_displacement),
where B ∮ dl is the magnitude of the magnetic field multiplied by the circumference of the loop.
Now, let's go through the calculation step by step:
1. Determine the circumference of the loop. Since the wire is infinitely long, you can assume a circular loop with a radius, r. Therefore, the circumference is given by 2πr.
2. Substitute the known values into the equation:
B × 2πr = μ₀(I_conduction + I_displacement).
3. Solve the equation for I_displacement:
I_displacement = (B × 2πr - μ₀I_conduction) / μ₀.
Now you can calculate the displacement current using this equation. Remember to convert all units to the appropriate SI units.
Regarding your second question, to compare the magnetic fields created by conduction current and displacement current, you are correct in using the formula E₀ × (dE/dt), where E₀ is the electric field and dE/dt is the rate of change of the electric field.
However, in this scenario, the question is specifically asking about the magnitude of the magnetic field. The magnitude of the magnetic field created by the displacement current, B_displacement, is related to the magnitude of the electric field and its rate of change as follows:
B_displacement = (μ₀/ε₀) × dE/dt,
where ε₀ is the electric constant or permittivity of free space (8.85 × 10^-12 C²/(N·m²)).
To determine the value of dE, you would need more information about the nature of the electric field changing during the cessation of the conduction current. Without this information, it is not possible to calculate the exact value of dE or make a comparison between the magnetic fields created by the conduction and displacement currents.