# A Problem regarding Charge density, some Calc mar by required

1. Jan 17, 2007

### paque

Hi,

i'm Having a bit of trouble with this challenge problem posed to us:

1. The problem statement, all variables and given/known data

A spherical cloud of charge of radius R contains a total charge of +Q with a nonuniform volume charge density that varies according to the equation:

$$\rho(r) = \rho_{0}(1- \frac{r}{R})$$
alt: p(r) = p0(1-(r/R))

for r <= R [meaning the charge is denser in the center]

and

$$\rho = 0$$
alt: p = 0

when r>R [outside of radius, R, there is no charge.]

where $$/rho$$alt: p is charge density

and r represents the distance from the center of the sphere,
and R represents the radius of the sphere itself

Algebraic Answers must be in terms of Q, R, and constants

(a) Determine the following as a function of r when r > R
i. The Magnitude, E of the electric field​

(b) A proton is placed at point P away from the sphere is released. Describe its motion for a while after its release.

(c) derive an expression for p0 [rho sub zero] in the p(r) equation

(d) Determine the magnitude, E of the electric field as a function of r for r <= R

EDIT: I found a copy of the problem online: http://www.collegeboard.com/prod_downloads/ap/students/physics/physics_c_em_frq_03.pdf (first of the free response problems)

2. Relevant equations

Of course the equations for a sphere would be pertinent:

Volume = (4/3)pi * r^3
and
Surface Area = 4pi * r^2

and i have learned http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html" [Broken]:

Force, F=K*Q*q / R^2; K = 9E9 N*m^2 * C^-2

alt: F= (1/(4 * pi * E)) * (Q*q/R^2)

and http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html" [Broken]:

phi = Q/E

recently.

3. The attempt at a solution

(a).i could wish that i could treat the sphere as a point charge, with a net charge of +Q...
so Magnitude of a field, E = kQ/r^2

but, if no such luck, i was thinking that, perhaps, some calculus may be required:

http://img149.imageshack.us/img149/7008/qcharge1wc8.th.png [Broken]
or something.

[perhaps i did [c] by accident... ?]

: since the sphere has a positive charge, obviously, the photon moves away from the sphere, ever accelarating, due to the force from the sphere, but accelarating less and less.

[c]: i sincerely haven't a clue.... i can barely comprehend what p0 [rho sub zero] represents in the equation

(d): i think that this is similar to [a], except that instead of big R, you'd submit, r

generally speaking... i'm not really up to scratch with my calculus, and this problem is somewhat difficult for me due to my lack of comprehension...

and help at all would be greatly appreciated...

thankyou, Daniel: divine.path@gmail.com

Last edited by a moderator: May 2, 2017
2. Jan 18, 2007

### tim_lou

1. imagine a gaussian surface (a sphere), where the point you want to evaluation the field strength is on the sphere.

2. find the enclosed charge by integration. this should be constant as long as r>R
3. how can you find the field strength by gauss's law?

For a) by Gauss's law, in what circumstance, can you treat a charge distribution as a point charge?

what happens when r<R? how would the enclosed charge change?

once you get the E as a function of r, all the other parts easily follow.