# Gauss's law (2)

1) A shperical sheel has inner radius R_in and outer radius R_out. The shell contains total charge Q, uniformly distributed. The interior of the shell is empty of matter and charge. Find the electric field for (a)r>R_out, (b)r<R_in, and (c)R_in<r<R_out respectively. Then sketch a graph of E versus r. [E=electric field]

These are the answers I get:
a) E=KQ/r^2 [radially outward] where K=9.0x10^9
b) E=0
c) [KQ(r^3-R_in^3)]/[r^2(R_out^3-R_in^3)] where K=9.0x10^9

My problem is for the sketching part on the interval R_in<r<R_out. How can I know the shape of [KQ(r^3-R_in^3)]/[r^2(R_out^3-R_in^3)]. Can I just do some cancellation like r^3/r^2 = r? Would it just be a straight line? But the expression seems like it's a rational function, so would it still be linear?

However, to sketch rational functions would require 3 full pages of analysis using calculus...is there any way to do a quick sketch for this part while getting the correct shape?

Does anyone have any idea?
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Tom Mattson
Staff Emeritus
Gold Member
How about using Maple? Assume convenient values for $R_{out}$ and $R_{in}$ (like 2 and 1, respectively). Then you can get an idea of the shape of the graph.

What is maple?

Would a rational function with a degree of 3 in the numerator and a degree of 2 in the denominator always gives something that is roughly linear for R_in<r<R_out ? I tried using some graphing software to graph it, and it seems quite linear for R_in<r<R_out
[but I don't know how to figure the shape out without a graphing software]

Tom Mattson
Staff Emeritus