B field due to a wire with an alternating current

[AxiomOfChoice]

Homework Statement

I need to find the magnetic field a distance $r$ from a long, thin wire carrying a current $I(t) = I_0 \sin \omega t$.

Homework Equations

Field a distance $r$ from a wire carrying a steady current $I$ in the $z$ direction:

$$\vec B(r) = \frac{\mu_0 I}{2 \pi r} \hat \phi$$

The Attempt at a Solution

I'm tempted to say that, in the case of the alternating current,

$$\vec B(r) = \frac{\mu_0 I_0 \sin \omega t}{2 \pi r} \hat \phi,$$

but I'm not sure I'm right, and I certainly can't explain to myself why it should be Ok to assume that the DC result will be the same as the AC result. My HW problem is actually very complicated: There's a lot of stuff involving induced EMF and time-average power. But I think I'm golden on this problem if I can just figure out what the $\vec B$-field should be. Thanks!

 

[Delphi51]

I think you are right. Inductance effects will cause the AC potential across the wire to be different from the DC, but you are given current and it directly causes the B field so it shouldn't matter whether DC or AC.

 

[kerimek]

I would first clarify that the equation for the magnetic field due to a steady current in a wire, \vec B(r) = \frac{\mu_0 I}{2 \pi r} \hat \phi, assumes that the current is constant and not changing over time. In the case of an alternating current, the current is continuously changing, so the equation for the magnetic field would also need to take into account this time-varying aspect.

One way to approach this problem would be to use Maxwell's equations, specifically the Ampere-Maxwell law, which states that the curl of the magnetic field is equal to the current density. In this case, the current density is changing over time due to the alternating current, so the magnetic field would also be changing over time.

Another approach would be to use the Biot-Savart law, which describes the magnetic field due to a small element of current. This law can be applied to each small element of the wire, taking into account the changing current over time, to calculate the total magnetic field at a distance r from the wire.

In either case, it is important to keep in mind that the magnetic field will not have a constant magnitude or direction due to the changing current, and will instead vary over time. It is also important to consider the direction of the magnetic field, as it will not necessarily be in the same direction as the current, but will depend on the geometry of the wire and the direction of the current.

In summary, the equation you have proposed, \vec B(r) = \frac{\mu_0 I_0 \sin \omega t}{2 \pi r} \hat \phi, is not a complete or accurate representation of the magnetic field due to an alternating current in a wire. Further analysis and consideration of the time-varying aspect of the current is necessary to accurately describe the magnetic field in this situation.