Average electric field over a spherical surface

In summary, the conversation is about working out problem 4 in chapter 3 of Introduction to Electrodynamics by Griffiths. The problem involves finding the average electric field over a spherical surface due to charges outside the sphere and inside the sphere. The conversation also mentions a figure by Griffiths and questions the use of the law of cosines in the solution.
  • #1
JD_PM
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Homework Statement


I was working out problem 4, chapter 3 of Introduction to Electrodynamics by Griffiths:

a) Show that the average electric field over a spherical surface, due to charges outside the sphere, is the same as the field at the centre.

b) What is the average due to charges inside the sphere?

The Attempt at a Solution



I know the solution, but there are a few aspects I do not see:

Screenshot (93).png

Screenshot (94).png

Here you have the image Griffiths makes reference at:

Screenshot (95).png


Why ##z^2 + r^2 - R^2 = z - Rcos \theta## (law of cosines developement)?
 

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  • #2
JD_PM said:
Why ##z^2 + r^2 - R^2 = z - Rcos \theta## (law of cosines developement)?
This equality is not correct. Instead, note the following substitution
upload_2019-2-19_13-44-35.png
 

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Related to Average electric field over a spherical surface

1. What is the formula for calculating the average electric field over a spherical surface?

The formula for calculating the average electric field over a spherical surface is E = Q/4πε0R2, where Q is the total charge enclosed within the surface, ε0 is the permittivity of free space, and R is the radius of the spherical surface.

2. How does the average electric field over a spherical surface differ from the electric field at a specific point on the surface?

The average electric field over a spherical surface takes into account the entire surface and the charge enclosed within it, while the electric field at a specific point only considers the charge at that point. The average electric field is a measure of the overall effect of the charge distribution on the surface, while the electric field at a specific point is a measure of the force experienced by a charge placed at that point.

3. Can the average electric field over a spherical surface be negative?

Yes, the average electric field over a spherical surface can be negative if the charge enclosed within the surface is negative. This would indicate an attractive force towards the center of the spherical surface.

4. How does the average electric field over a spherical surface change with distance from the center?

The average electric field over a spherical surface follows an inverse square law, meaning that it decreases as the distance from the center increases. This is because the surface area of the spherical surface increases with distance, spreading out the same amount of charge over a larger area.

5. What is the significance of calculating the average electric field over a spherical surface?

Calculating the average electric field over a spherical surface allows for a better understanding of the overall effect of a charge distribution on the surface. It can also be used to determine the electric flux through the surface, which is an important concept in electromagnetism.

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