How Is the Total Energy Inside a Cylindrical Wire Calculated?

[Jesssa]

Given a cylindrical wire of radius r, length L, carrying a current I, find the total energy stored inside the wire.

From griffiths,

u_em= εE²/2 +B²/2μ

and the tot energy is

∫u_em dVI have my E and B fields, but my B field is a function of x where x<r, (E is uniform)

B=kx/r² (k=all the constants)

my question is,

it says inside the wire, does this mean i cannot put x=r and integrate easily to get

Energy=(εE²/2 +B²/2μ)$\pi r^2 L$ ?

will i have to integrate B separately to get something like

∫∫∫Kx x dx dz d$\phi$ where K = k/r² (since dV = x dx dz dthi in cylindrical)

= (2/3) K x³ L pi

If so would this be it? No bounds on x, giving the energy at some radius inside the wire?

I guess the real question is, does saying INSIDE mean not evaluated at the boundary? Like the total energy inside the wire at any radius x<r

Im unsure about this because of the (2/3)x³ in the second approach since if you put x=r here it will be different to the first approach because of the (2/3)

 

[tiny-tim]

Hi Jesssa!

Given a cylindrical wire of radius r, length L, carrying a current I, find the total energy stored inside the wire.
…
I guess the real question is, does saying INSIDE mean not evaluated at the boundary? Like the total energy inside the wire at any radius x<r

The total energy stored inside the wire is the the energy per tiny volume, integrated over the whole volume.

 

[Jesssa]

hey tiny-tim,

the energy density is u_em and the total energy is the integral of this over the whole volume, this is straight from griffiths, but I'm not sure about the solution to the problem,

do you know which of the two cases i posted are the correct approach?

the first is taking B at the surface B(r)=k/r = K

then the total energy is just what i posted in the first post,

(εE²/2 +B²/2μ)πr²L

and the second was leaving B as a function of x, the field at some distance x from the centre of the circular cross section,

B = kx/r²

integrating over the volume in cylindrical co-ordinates to get

∫∫∫Kx x dx dz dϕ where K = k/r² (since dV = x dx dz dthi in cylindrical)

= (2/3) K x³ L pi i guess the question is over what volume is considered inside the cylinder, some x<a or just x=a?

or was your post implying that neither approaches are correct?