- #1
pmb_phy
- 2,952
- 1
In "Classical Electrodynamics - 3rd Ed.," J.D. Jackson has an exercise, 1.10, to derive the mean value theorem of electrostatics. Does anyone know of a derivation which is located on the web?
Pete
Pete
No. It isn't. Itsdextercioby said:"The average value of the potential over the spherical surface is
[tex] \bar{\Phi} =\frac{1}{4\pi R}\int \Phi \ dA [/tex]
pmb_phy said:No. It isn't. Its
[tex] \bar{\Phi} =\frac{1}{A}\int d\Phi [/tex]
where A is the area of the sphere.
Pete
dextercobi said:[tex] \bar{\Phi} =\frac{1}{4\pi R}\int \Phi \ dA [/tex]
That would be true if you were talking about a line integral rather than in this case where the integral is a surface integral. What [itex]d\Phi[/itex] is? Hmmm. Perhaps you're right. I can't say that is a small element of [itex]\Phi[/itex] since that makes no sense to me at the moment. Thanks.qbert said:This isn't quite right either.
It isn't clear what you mean by [itex] \int d\Phi [/itex]
since usually the integral of an exact differential
is just the function evaluated at the endpoints.
May I propose a more [notationally] precise formula:qbert said:I propose another formula:
[tex]
\bar{\Phi} = \frac{1}{4 \pi R^2} \int_A \Phi dA
[/tex]
This is a Homework subforum... and, in my experience, my students appreciate seeing the general idea in a consistent memorable notation rather than a special case. After all, there was some confusion with this formula.qbert said:Ah, pedantry, thou most noble virtue. You are, of course, correct. Now the next person (who reads this thread) who needs a 2-D average will know the correct generalization.
Hi Robrobphy said:This is a Homework subforum... and, in my experience, my students appreciate seeing the general idea in a consistent memorable notation rather than a special case. After all, there was some confusion with this formula.
Although the notation "A" [implicitly] suggests an area average, it is certainly not necessarily so.
The Mean Value Theorem of Electrostatics is a mathematical theorem that states that at any given point in an electrostatic field, there exists at least one point where the electric potential is equal to the average of the potential at all points on a closed surface surrounding that point.
The Mean Value Theorem is important in electrostatics because it allows us to make predictions about the behavior of electric fields. It helps us understand the relationship between the potential at a specific point and the overall potential of the entire field.
The Mean Value Theorem can be used to solve various problems in electrostatics, such as calculating the electric potential at a specific point in a field or determining the average potential on a conducting surface.
One limitation of the Mean Value Theorem is that it only applies to electrostatic fields that are continuous and have a finite number of singularities. Additionally, it assumes that the field is conservative, meaning that work done in moving a charge is independent of the path taken.
The Mean Value Theorem is closely related to other concepts in electrostatics, such as the concept of equipotential surfaces. It also has connections to Gauss's Law, which relates the electric field to the charge distribution within a closed surface.