Showing Infinite Self-Inductance of a Long Wire

[leoflindall]

Homework Statement

Show that the calculated self-inductance of a long straight wire of length l and outer
radius R becomes infinite when Maxwell’s extra term in the generalisation ofAmpere’s
law is ignored.

Homework Equations

(intergral over a closed curve) = meu0.I + meu0.(permitivity of free space) (d (magnetic flux))/dt

The Attempt at a Solution

I am very stuck on this. i considered the magnetic field flux both inside and outside the wire. Each magnetic field line must contribute to the total flux, but only once., Is this possibly to do with the two independant flux's interaction with each other?

Id appreciate any hints or help in moving me the right way!

 

[Jhero]

Thank you for your post. This is an interesting problem and I would be happy to help you with it.

First, let's start with the basics. The self-inductance of a wire is defined as the ability of the wire to produce an induced voltage in itself due to a change in current. It is represented by the symbol L and is measured in henries (H).

When considering a long straight wire, we can use Ampere's law to calculate the magnetic field produced by the wire. Ampere's law states that the integral of the magnetic field around a closed loop is equal to the permeability of free space (μ0) times the current passing through the loop.

However, Maxwell's correction to Ampere's law takes into account the displacement current, which is the changing electric field that can create a magnetic field. This correction is represented by the term μ0ε0(dΦE/dt), where ε0 is the permittivity of free space and ΦE is the electric flux passing through the loop.

Now, when we consider a long straight wire, the electric flux passing through the loop is zero since the electric field is perpendicular to the loop. This means that the extra term in Maxwell's correction becomes zero and we are left with just the original Ampere's law.

However, if we ignore this extra term and use only Ampere's law, we are neglecting the changing electric field and the resulting magnetic field. This leads to an infinite self-inductance because without the displacement current, there is no opposing magnetic field to counteract the changing current in the wire.

I hope this explanation helps you understand why the calculated self-inductance becomes infinite when Maxwell's extra term is ignored. Please let me know if you have any further questions.