The first thing I'll do is find how much interstellar matter likely exists in a cylinder 100km in diameter and 10 light-years long.
$V=\pi r^2 h$ is the volume of a cylinder, so let's find our volume in uh cubic centimeters:
$\pi × (110^{7})^2 × 9.46110^{18} = 2.97210^{26}cm^3$
Now let's find its mass.
Hydrogen has a mass of about $1.67410^{-24} g$ and helium about $6.64610^{-24} g$
Interstellar particle density is about 1 atom per cubic centimeter, that atom being 75 percent of the time hydrogen and about 25 percent the time helium. Let's get an average mass: $1.67410^{-24}(0.75) + 6.64610^{-24}(0.25) = 2.91710^{-24} g$
Our volume calculation is in cubic centimeters, and we have 1 particle per cubic centimeter, so there are about $2.97210^{26}$ particles inside out cylinder, each with an average mass of $2.91710^{-24} g$. 921
Finding total mass, we multiple the number of particles by the average mass and find that the total mass is about: $8.670*10^{2} g$, or about 867 grams of stuff. That's about a basketball and a couple of baseballs worth of mass.
So, basically nothing when compared to the stuff the beam will grab from within Jupiter.
If we assume Jupiter has an average density of $1.326 g/cm^3$, has a radius of $6.99110^{9}cm$, and the beam passes through say 60 degrees latitude in the south:
$\cos (60) × 2 × 6.99110^{9} = 3.49610^{9} cm$ is the length of that part of the beam which crosses Jupiter.
That part would have a volume of roughly: $\pi × (110^{7})^2 × 3.49610^{9} = 1.09810^{24} cm^3$
With an average density of $1.326 g/cm^3$, we can find mass to be: $1.456*10^{24} g$
That's a mass comparable to the dwarf planet Ceres.
As for what it'll do when it reaches the middle of nowhere in interstellar space 6 light-years away? I imagine that--being under the crushing pressure of Jupiter's interior no longer--it'll expand rapidly and magnificently, glowing hot at first before cooling down as its volume increases and heat escapes in greater quantities into cold, dark space. I don't think the gases' relative motion from its activities in the solar system will make much of a difference at all. 24