Is There a Displacement Current in a Wire with Increasing Current?

[Jamie_Pi]

Homework Statement

A wire with conductivity σ carries current I. The current is increasing at the rate dI/dt. Show that there is a displacement current in the wire equal to e₀/σ⋅dI/dt

Homework Equations

I_d = e₀⋅dφ/dt
dφ/dt = dE/dt ⋅ A (This is usually true, I'm not sure if it's useful in this case)
dV/dt=dI/dt⋅R (A changed version of Ohm's law)
R=1/σ
E=V/l

The Attempt at a Solution

I started by saying that dV/dt = dI/dt⋅1/σ, in which case the equation might look like:
I_d = e₀/(σ⋅l)⋅dI/dt⋅A

This looks like I'm going in the right direction, but I'm not sure how to get rid of l (because I don't know the length of the wire, and presumably it doesn't matter) or A (I think that the area of a wire is supposed to be nominal). Any tips that you have would be super great!

 

[Jamie_Pi]

I'm trying a new formula, but I've come to a point of stagnation:

dφ/dt = dI/dt ⋅ 1/σ

φ has units vm
so dφ/dt must have units vm/s
1/σ is the same thing as Ω, which has units v/A
and dI/dt has units A/s
vm/s = A/s⋅ v/A

the only problem I have now is that unit of length, which I don't know how to get rid of. I'm not sure what it represents. Any ideas?

 

[gleem]

1/σ is the same thing as Ω, which has units v/A

Doesn't σ have units of mhos/m while Ω has units of mhos^-1 or reciprocally σ ( 1/ohms⋅m ) while Ω (ohms)?

 

[Jamie_Pi]

Doesn't σ have units of mhos/m while Ω has units of mhos^-1 or reciprocally σ ( 1/ohms⋅m ) while Ω (ohms)?

Oh, I think that you're right. That's good, that just clears up the whole thing then.

 

[haruspex]

Doesn't σ have units of mhos/m ... or reciprocally σ ( 1/ohms⋅m )

You mean mho.m and m/ohm, right?

 

[gleem]

The unit of conductivity is mhos/m. A mho is 1/ohms so conductivity is 1/ohm⋅m.. Resistivity ρ =1/σ and has units of ohm⋅m or m/mho.

 

[haruspex]

The unit of conductivity is mhos/m. A mho is 1/ohms so conductivity is 1/ohm⋅m.. Resistivity ρ =1/σ and has units of ohm⋅m or m/mho.

Sorry, my mistake. I was thinking of resistance of a wire as being so many ohms per metre, but that's not what resistivity means; it's a bulk property..