Magnetic Field due to Time Dependent Current

In summary, the problem asks to find the magnitude of the magnetic field at a distance r from the centre of a long, straight copper wire with circular cross section and current sin(ωt). Using Ampere's Law and assuming a quasi-static approximation, the magnetic field can be calculated as B=μI/2πr. This approximation assumes a relationship between ω, ε and ρ and assumes that copper is a good conductor.
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Homework Statement


A long, straight, copper wire has a circular cross section with radius R, resistivity p and permittivity ε. If the current through the wire at any time t is sin(ωt) amperes, find the magnitude of the magnetic field B at time t a distance r from the centre of the wire for r > R.


Homework Equations


Ampere's Law:
μI = Bdl
Possibly Law of Biot-Savart:
B = μ/4π * (Idl x r)/r^2

The Attempt at a Solution


μI = Bdl
μ(sin(ωt))=B*dl (Since they are parallel)
μ(sin(ωt))=Bdl (Since B is constant radially around conductor)

This is where I reach a bottleneck. I don't know how to incorporate ε (since this is a magnetic field). I assume resistivity would be used in calculating the current I, but I don't know how that ties into the sinusoidal function.
 
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  • #2
maybe you don't need to use ##\epsilon## and ##\rho##. Also, you have already used a certain kind of assumption about ##\epsilon## and ##\rho## to get your answer. So maybe the question is hoping that you will explicitly state this assumption about ##\epsilon## and ##\rho##.

Your answer is essentially a kind of quasi-static approximation. (Not the most general answer for this question). But you have implicitly used the fact that copper is a good conductor, and assumed a certain relationship between ##\omega##, ##\epsilon## and ##\rho##. I'm not sure if that was your deliberate intention, or if you missed a few steps. But I think you have the answer they were looking for, but maybe without explaining under what approximation this answer will work.
 

1. What is a time-dependent current?

A time-dependent current is an electric current that changes over time. This can occur in various ways, such as through the use of alternating current (AC) in power systems or through the movement of charged particles in a conductor.

2. How does a time-dependent current create a magnetic field?

A time-dependent current creates a magnetic field through the movement of charged particles. As the current changes over time, the charged particles also move, which creates a changing magnetic field around the conductor.

3. What is the relationship between the strength of the magnetic field and the rate of change of the current?

The strength of the magnetic field is directly proportional to the rate of change of the current. This means that the stronger the current changes, the stronger the magnetic field will be.

4. How is the direction of the magnetic field determined by a time-dependent current?

The direction of the magnetic field is determined by the right-hand rule, which states that if you point your right thumb in the direction of the current flow, your fingers will curl in the direction of the magnetic field.

5. What are some applications of time-dependent currents and their magnetic fields?

Time-dependent currents and their magnetic fields have various applications in everyday life, such as in power generation and transmission, electric motors, and medical imaging devices. They are also used in technologies like magnetic levitation for high-speed trains and in particle accelerators for scientific research.

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