Electric field due to a surface charge on a sphere

[Bowtime]

I have been given a problem relating to surface charge distributions, however the surface is not an infinite plane, as i have found lots of examples for, it is for a sphere. the problem goes ' Consider a sphere of radius R which supports a uniform charge distribution of Rho(s) = 10 micro Coulombs/metre squared on its surface. determine the electric field produced by applying:

i) direct integration
ii) Gauss' law

determine the electric potential everywhere. Sketch the electric field and potential as functions of radial distance from the centre of the sphere.'



2. Has anyone seen a worked example anywhere for a question like this? I've seen plenty for an infinite plane, or for a sphere of uniform charge distribution but can't find anything for this question. can anyone point me in the right direction?



3. I know that for a plane the equation is E = Rho(s)/(2Eo) where Eo is permittivity of free space but really don't know where to start with this.

 

[grey_earl]

Hi Bowtime! Welcome to Physics Forums :)

Assuming you mean the equation
$$\phi(\vec{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\rho(\vec{r}')}{\lvert \vec{r}-\vec{r}' \rvert} \mathrm{d}^3 x$$
with "direct integration", for a spherical surface charge distribution ρ is given by
$$\rho(\vec{r}) = \rho_0 \delta(\lvert \vec{r}-\vec{r}_0 \rvert - R)$$
where ρ0 is the surface charge density, r0 is the position of the sphere's center and R is its radius.

Have you already worked with δ distributions? They are usually taught in electrodynamics courses, but if not, just ask again.

 

[Bowtime]

Hi grey-earl,

Thanks for the response. I'm doing a course in Power Systems engineering and this came up in the Electrostatic and Electromagnetic field theory module. It's the first time I've come across this and I'm one of those people that learns best from worked examples. I've scoured the web and several textbooks but can't find anything other than surface charge examples for flat planes, thanks for the equation if i can find that i'll be on the right path :)

 

[Bowtime]

In case this applies to anyone else i found an example here: [URL="http://ocw.mit.edu/courses/physics/8-02sc-physics-ii-electricity-and-magnetism-fall-2010/conductors-and-insulators-conductors-as-shields/MIT8_02SC_notes9.pdf"[/URL]. i think this is what I've been looking for.

 

[SammyS]

In case this applies to anyone else i found an example here: http://ocw.mit.edu/courses/physics/8-02sc-physics-ii-electricity-and-magnetism-fall-2010/conductors-and-insulators-conductors-as-shields/MIT8_02SC_notes9.pdf". I think this is what I've been looking for.

That link covers Gauss's Law, but you're also supposed to find the electric field by direct integration (I presume, by applying Coulomb's Law).

 

[KanedaSyndrom]

That link covers Gauss's Law, but you're also supposed to find the electric field by direct integration (I presume, by applying Coulomb's Law).

I have this exact problem in "Electromagnetics for Engineers" by Clayton, R. Paul.

If anyone else is in possession of this textbook, I'm having trouble with review exercise 3.3 and presumable 3.4 as well once I get to that one. The previous examples only covered a line charge and an infinite surface charge plane.

In exercise 3.3 I'm supposed to find the electric field for a spherical surface charge distribution with radius a, for all r > a.

I'm 100% sure it's supposed to be solved via Coulomb's Law in this exercise.

The exact exercise text is as follows:

Determine the electric field due to a sphere of charge of radius a for r>a where the charge is uniformly distributed over the sphere surface with distribution ps C/m^2 (hint: place the sphere at the origin of a spherical coordinate system)