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Homework Statement
A) use gauss's Law to determine the electric field at all values of radial distance (0<r<infinity) from the center of a nonuniformly charged cylinder that is very very long and lies along the xaxis. The cylinder carries excess charge per volume ρ=[a]r^2 (the [] are supposed to be absolute value) and has total radius,R.
B) a nonconductive sheet of paper lies in the xzplane such that the bottom edge (width) of the paper infinitesimally close to the edge of the cylinder from part (A) and has its width parallel to the axis of the cylinder. The paper has width,W, along the xaxis, and length,L, along the zaxis, and carries a nonuniform linear charge density ([z]R) (again the [] represent abs value). What is the total force acting on the sheet of paper in terms of the given constants (a,b,R,w,ε_{0} and L)
The problem is attached(IMG_4007. JPG) below.
Homework Equations
gauss's Law
The Attempt at a Solution
I have uploaded some of my interpretation of the problem (My interpretation.jpg)
My first thought is to look at a small segment of the cylinder with radius, r, thickness, dr, and charge, dQ.
we will find dQ by:
dQ=ρdV
dV=2πrLdr
dQ=ρ2πrLdr, however it is not uniform charge density. the problem states the cylinder carries excess charge per volume of ρ=ar^{2} (the "a" is in abs value)
so
dQ=ar^{2}2πrLdr
when we integrate dQ we will get Q_{enclosed} within our gaussian surface.
Limits of integration for below integral are 0→R_{1} (R_{1} is the radius of my gaussian cylinder)
Q=∫dQ=∫ar^{2}2πrLdr=a2πL∫r^{3}dr→→(aπLR_{1}^{4})/2
we now have our Q enclosed for gauss's Law
skipping a few steps of writing out gauss's law
E2πR_{1}L=(aπLR_{1}^{4})/(2ε_{0})→ E=(aR_{1}^{3})/(4ε_{0})
For outside the cylinder we basically change limits when integrating dQ, Outer Gaussian cylinder radius R_{2}
so our new limits of integration are 0→R
Qenclosed=a2πL∫r^{3}d→→(aπLR^{4})/2
plugging into gauss's Law
E2πR_{2}L=(aπLR^{4})/(2ε_{0})→E=(aR^{4})/(4R_{2}ε_{0})
Now for part B.
I was talking with @TSny about a similar problem the other day, except this has a nonuniform charge density...I wasnt 100% good with the one with a uniform charge density. I need things broken down very basically. Not asking to give me the answer (as that is against the rules!!!), but definitely need some help.
Thanks in advanced, you guys/girls have been awesome so far!!!!!!!!!!!!!! :)
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