E field calculation for q sphere

fisher garry · Dec 7, 2019

Textbook problem 2.7: find the electric field a distance z from the center of a spherical surface of radius R with uniform charge density sigma, for z < R and z > R, in terms of total charge q. Hint uses law of cosines with sqrt(R^2 + z^2 - 2Rz) equal to R - z if R > z, else z - R.

Worked solution to Problem 2.7 finding the electric field of a uniformly charged spherical shell, with a diagram of a sphere of radius R and an observation point at height z on the axis; the final result shows E equal to q divided by 4 pi epsilon zero z squared outside, and zero inside.

I have some questions about this answer. Why do they use absolute value when writing in the limits in the integral underlined with orange?
And how do they get from this value where I have underlined with orange to the answer for E outside the sphere. Can someone do the rewriting?
And last why is the E field 0 inside the sphere? Is it related to the calculations or not? Can anyone explain why it is 0?

I am using this to try to derive gauss law without the divergence theorem

berkeman · Dec 7, 2019

Is there a reason that your attached image is so fuzzy? Can you link to the source?

fisher garry · Dec 7, 2019

I have scanned Griffiths introduction to electrodynamics onto my computer (I have bought two copies of the book). I marked the equality sign in the question. What else is unclear? Is it the question or the answer?

berkeman · Dec 7, 2019

fisher garry said:

What else is unclear?

The image is extremely fuzzy, and it is problematic for you to ask us to squint our eyes to try to decode it. Please post a much clearer scan or image. Thank you.

fisher garry · Dec 7, 2019

here is a different answer. Albeit my questions are in thefirst image in my original post. But this is the same assignment

Printed textbook page titled Example 2 deriving the electric field of a spherical shell of charge, with equations 8 to 12 including the cosine law distance, the cos alpha expression, and the integral for E outside the sphere.

Printed textbook page with equations: since 4πR²σ is the total charge, the sphere acts as a point charge outside; for z < R inside the shell, E = 0 (eq. 13), so a uniformly charged spherical shell produces no field inside.

E field calculation for q sphere

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