Electric Field of a Spherical Shell: Sketching E vs. r

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For a spherical shell with inner radius R_in and outer radius R_out containing a total charge Q, the electric field is defined for three regions: outside the shell (r > R_out) where E = KQ/r^2, inside the shell (r < R_in) where E = 0, and within the shell (R_in < r < R_out) where E is given by a rational function. The challenge lies in sketching the electric field for the region R_in < r < R_out, as the expression appears complex. Using graphing software or tools like Maple can help visualize the shape of the graph, which tends to be linear in this interval due to the degrees of the numerator and denominator. Understanding the behavior of rational functions can simplify the sketching process without extensive calculus analysis.
kingwinner
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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)][/color] 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?
Thanks for helping!:smile:
 
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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]
 
Maple is a computer algebra system that has graphing capabilities. It's available for free at just about every college and university in North America. But if you don't know what Maple is, then you could use a graphing calculator.
 
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