A Problem regarding Charge density, some Calc mar by required

In summary, the conversation is discussing a challenge problem involving a spherical cloud of charge with a nonuniform volume charge density. The goal is to determine the electric field magnitude as a function of distance from the center of the sphere, the motion of a proton placed near the sphere, an expression for the charge density constant, and the electric field magnitude for distances smaller than the radius of the sphere. The suggested approach involves using Gauss's law and integrating to find the enclosed charge. The conversation also mentions the use of equations for volume, surface area, and force due to a point charge.
  • #1
paque
4
0
Hi,

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


Homework Statement



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:

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

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

and

[tex]\rho = 0[/tex]
alt: p = 0

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

where [tex]/rho[/tex]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)

Homework 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.

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 accelerating 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
 
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  • #2
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.
 
  • #3


Hello Daniel,

Thank you for reaching out about this problem. I can understand how challenging it can be to tackle complex problems like this, especially when they involve calculus.

To start, let's break down the problem into its different parts and see if we can find some solutions.

(a) To determine the magnitude of the electric field as a function of r when r > R, we can use Gauss's Law. This law states that the flux of the electric field through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space (ε0). In this case, the closed surface can be a sphere of radius r, and the charge enclosed would be Q. So, we can write:

Φ = E * 4πr^2 = Q/ε0

Solving for E, we get:

E = Q/(4πε0r^2)

However, this equation is only valid for r < R, since for r > R there is no charge enclosed by the sphere. So, we can write the final expression for the electric field as:

E = Q/(4πε0r^2) for r < R
E = 0 for r > R

(b) For this part, we need to use Newton's second law (F = ma) to describe the motion of the proton. Since we know the force acting on the proton (from part a), we can write:

F = ma = kQq/r^2

Where q is the charge of the proton and k is the Coulomb constant. This equation tells us that the proton will experience a force towards the sphere, causing it to accelerate. As it moves closer to the sphere, the force will increase, causing the proton to accelerate more and more. Eventually, it will reach a point where the force is balanced by the proton's inertia, and it will start to decelerate. This process will continue until the proton comes to a stop at the surface of the sphere. After that, the proton will start to move in the opposite direction, away from the sphere, due to the repulsion of like charges.

(c) To determine an expression for p0 in the p(r) equation, we can use the fact that the total charge of the sphere is Q. So, we can write:

Q = ∫ρ(r) dV

Where dV is a small volume element and the integral
 

1. What is charge density?

Charge density is a measure of the electric charge per unit volume of a substance. It is typically denoted by the symbol ρ and has units of coulombs per cubic meter (C/m³).

2. How is charge density calculated?

To calculate charge density, you divide the total electric charge of a substance by its volume. The formula for charge density is ρ = Q/V, where Q is the total charge and V is the volume.

3. What is the significance of charge density?

Charge density plays a crucial role in understanding the behavior of electric fields and the interactions between charged particles. It is also used in various applications such as in capacitors and in determining the electrical properties of materials.

4. What is the relationship between charge density and electric field?

The electric field is directly proportional to the charge density. This means that an increase in charge density will result in an increase in the strength of the electric field. Conversely, a decrease in charge density will result in a weaker electric field.

5. What is the connection between charge density and Coulomb's law?

Coulomb's law states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. Charge density is a factor in this equation, as it determines the amount of charge within a given volume and therefore affects the strength of the force between charged particles.

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