Two oppositely charged wires and their capacitance

[mooncrater]

Homework Statement

Two long wires with the same cross section are arranged in air PARALLEL to each other . Assume both have opposite charge densities +λ and -λ. The distance between the axis of the wire is η times larger than the radius(r) of wires cross section . What would be the capacitance of the wires per unit lenth?

Homework Equations

Electric field of any point between these wires at a distance x from the +λ charged wire is
λ/2πε[1/(x)+1/(ηr-x)]

The Attempt at a Solution

The problem I am facing is about the limits to be used while integrating Electric field to get the potential .
We know that V=-∫Edx
(Between some limits)
So ∫Edx =λ/2πε[ln(x/ηr-x)]
And if I integrate it over 0 to ηr
The I will get stuff like ln 0 and ln∞ which are obviously not defined . Some other limits just don't get the right answer which is πε/ ln η . [/B]

 

[Tanya Sharma]

Homework Equations

Electric field of any point between these wires at a distance x from the +λ charged wire is λ/2πε[1/(x)+1/(ηr-x)]

1) 'x' gives distance between which two points ?
2) What is the initial point ?
3) Now answer ,what should be the lower limit in the integration ?

 

[mooncrater]

O-----------O
<----->is x
Initial point is the centre of the first+λ charged wire (I think).
So what I think is that the lower limit should be 0 and the upper limit should be ηr. BUT , when I put them in the integrated electric field the answer is absurd...

 

[Tanya Sharma]

O-----------O
<----->is x
Initial point is the centre of the first+λ charged wire (I think).

Right .

But , does electric field exist within the wire ? The electric field does work from the surface of wire +λ to the closest point on the surface of other wire -λ.

 

[mooncrater]

No. There is no electric field inside the wires ... therefore I should take the limits from r to ηr-r.
After doing the calculation I found that the potential is
V=λ/πε[ln(η-1)]
Since we're talking about a unit length thus λ=Q
So Capacitance=C=Q/V=πε/ln(η-1)
Which is just a little different from the answer(which now seems incorrect to me) So am I right here?
Edit: hey !hey!hey! In the question its given that η>>1 thus η-1≈η so now I got the correct answer ...now I am happy... thank you for your guidance dd.!