Calculating Self-Inductance of Long Current-Carrying Wire

[iitjee10]

I was trying to calculate the self inductance (per unit length) of the following system using two methods:

System : A long current carrying wire of radius R carrying uniform current density and the same current returning along the surface. (Assuming that the surface is insulated by a very thin sheet).

Method 1 : I calculated B inside and found out flux. Then I used $$\Phi = LI$$
The answer came out to be $$\frac{\mu_{o}}{4\pi}$$

Method 2 : I calculated the energy associated with the magnetic field and equated it to $$\frac{1}{2}LI^{2}$$. The answer came out to be $$\frac{\mu_{o}}{8\pi}$$

Which one is correct and where is the mistake in the wrong one ?

 

[zorro]

Can you show your work for method 1?

 

[iitjee10]

Using Ampere's Circuital Law,
$$B_{inside}*2 \pi s = \mu_{o}\frac{I \pi s^{2}}{\pi a^{2}}$$

$$=> B_{inside} = \frac{\mu_{o}Is}{2\pi a^{2}}$$

$$=> d \Phi = B_{inside}.da = \frac{\mu_{o}Is}{2\pi a^{2}}dsdz$$

$$=> \Phi = \int B.da = \frac{\mu_{o}Il}{4 \pi} = LI$$

$$=> L_{per unit length} = \frac{\mu_{o}}{4 \pi}$$

 

[iitjee10]

anyone ??

 

[Doc Al]

Your Method #1 calculation looks OK. Show how you calculated the energy density for Method #2.

 

[ajith.mk91]

Hey Hi! I am not here to answer your question and sorry if that disappoints you but how do you define flux linkage in this particular situation?

 

[iitjee10]

For the energy method

$$U = \frac{1}{2\mu _{o}}\int B^{2}dV$$

$$=> U = \frac{1}{2\mu _{o}} \int \frac{\mu _{o}^{2}s^{2}}{4\pi ^{2}a^{4}}sdsd\phi dz$$

phi varies from 0 to 2pi, s from 0 to a and integral of dz = l

equating value of U with 0.5LI^2 we get
$$L_{perunitlength} = \frac{\mu _{o}}{8 \pi}$$

 

[cragar]

Should there be an I in your integral .

 

[John94N]

iitjee10,

Not an easy problem.
I think there is a problem in your third equation which says that flux is linearly proportional to z. Finding the inductance of a straight length of wire is complex. Here is a reference http://www.ee.scu.edu/eefac/healy/indwire.html. Good luck.

 

[zorro]

The answer you got through method 1 is wrong.
The correct answer is the one you got in your second method. Once I figure out why method 1 is wrong I will post it here.