What is the Electric Field and Potential of a Non-Uniformly Charged Sphere?

[joemama69]

Homework Statement

A non-uniformly charged sphere of radius R has a charge density p = p_o(r/R) where p_o is constant and r is the distrance from the center of the spere.

a) find the total charge inside the sphere

b) find the electric field everywhere (inside & outside sphere)

c) find the electric potential difference between teh center of the sphere and a distance r away from the center for both r<R and r>R.

d)Graph the potential as afunction of r

Homework Equations

The Attempt at a Solution

a) Find the total charge in sphere.

p = dQ/dV

Q = integral p dV where dV = 4(pi)R^2dR and it is integrated from 0 to R

Q = integral from 0 to R p_o*(r/R)*4*pi*(R^2)*dR = 2*pi*p_o*r*(R^2)

b) Find electric Field inside & out of sphere

Outisde r>R

Q = 2*pi*p_o*r*(R^2)

\ointEdA = Q_encl / E_o = E(4*pi*(R^2)) = (2*pi*p_o*r*(R^2))/E_o

E = (p_o*r)/2E_o

Inside... r<R

If the rest is correct, can i get a hint on finding the inside as to the outside. this confuses me. it seems the only difference is r<R, but how would that effect the equation.

 

[zachzach]

Homework Statement

A non-uniformly charged sphere of radius R has a charge density p = p_o(r/R) where p_o is constant and r is the distrance from the center of the spere.

a) find the total charge inside the sphere

b) find the electric field everywhere (inside & outside sphere)

c) find the electric potential difference between teh center of the sphere and a distance r away from the center for both r<R and r>R.

d)Graph the potential as afunction of r

Homework Equations

The Attempt at a Solution

a) Find the total charge in sphere.

p = dQ/dV

Q = integral p dV where dV = 4(pi)R^2dR and it is integrated from 0 to R

Q = integral from 0 to R p_o*(r/R)*4*pi*(R^2)*dR = 2*pi*p_o*r*(R^2)

b) Find electric Field inside & out of sphere

Outisde r>R

Q = 2*pi*p_o*r*(R^2)

\ointEdA = Q_encl / E_o = E(4*pi*(R^2)) = (2*pi*p_o*r*(R^2))/E_o

E = (p_o*r)/2E_o

Inside... r<R

If the rest is correct, can i get a hint on finding the inside as to the outside. this confuses me. it seems the only difference is r<R, but how would that effect the equation.

dQ = \rho dv = (\frac{\rho_0}{R})r 4\pi r^2 dr

So your integral for (a) is wrong because your dv is wrong. To find the E field inside the sphere draw your Gaussian surface at some arbitrary r inside the sphere. What is Q_{enc}? You know that dQ = \rho dv but what are your limits now?

 

[joemama69]

ok so for part A i integrated and got Q = (4\pip_or⁴)/R and wouldn't that be the same as Q_encl because it was the charge in the sphere

E_inside = (1/4\pir²)(Q_encl/E_o)

what differes when i find it outside

 

[zachzach]

ok so for part A i integrated and got Q = (4\pip_or⁴)/R and wouldn't that be the same as Q_encl because it was the charge in the sphere

E_inside = (1/4\pir²)(Q_encl/E_o)

what differes when i find it outside

Q_{total} = \frac{4\pi\rho_0}{R}\int_{0}^{R}r^3dr = <br /> \frac{4\pi\rho_0}{R}\frac{R^4}{4} = \pi\rho_0R^3

To find charge inside all you do is change your limits from 0 -> R to 0 -> r where r is an arbitrary radius inside the sphere.

p.s learn latex it's not that hard https://www.physicsforums.com/showthread.php?t=8997.

 

[zachzach]

So if you want the E field outside the sphere, Q_{enc} = Q_{total} since the whole sphere is enclosed with your Gaussian surface.

 

[joemama69]

ok so for part a i wanted the total charge inside sphere which would be Q_enc.. correct?

Part A)

Q_encl = \pip_or⁴/R

Then for part B i need the Electric field inside and outside

Inside..

E(4\pir²) = Q_encl/E_o

and then outside i would solve for Q_total as you did earlier and use that instead of Q_encl in the same equation for the inside


why does my pi always end up like an exponent

 

[zachzach]

ok so for part a i wanted the total charge inside sphere which would be Q_enc.. correct?

Part A)

Q_encl = \pip_or⁴/R

Then for part B i need the Electric field inside and outside

Inside..

E(4\pir²) = Q_encl/E_o

and then outside i would solve for Q_total as you did earlier and use that instead of Q_encl in the same equation for the insidewhy does my pi always end up like an exponent

Use latex and that link I gave you, it is simple.

(a) Your answer is wrong (I showed you the answer in a previous post) but you are right that Q_{enc}(r) = \frac{\pi\rho_0}{R}r^4 --> This equation is the enclosed charge as a function of your radius with r between 0 and R. It is asking for the total charge of the sphere though. So what is the total enclosed charge if the whole sphere is included? It is just that equation but with r = R.

(b) For inside the sphere: Looks good so far and you know what Q_{enc} is :
Q_{enc}(r) = \frac{\pi\rho_0}{R}r^4.

For outside the sphere: You know what Q_{enc} = Q_{total} or in otherwords your answer from (a).

 

[joemama69]

ok sorr i was confused.. i thought that the charge inside would not include the total sphere

ok now i need to find the potential difference for r<R and r>R
Im confused on the integration limits

r<R do i integrate from 0 to R

r>R i integrate from R to r

 

[zachzach]

ok sorr i was confused.. i thought that the charge inside would not include the total sphere

ok now i need to find the potential difference for r<R and r>R
Im confused on the integration limits

r<R do i integrate from 0 to R

r>R i integrate from R to r

It asks for the potential difference FROM THE CENTER OF THE SPHERE and a distance r away from the center

r<R is that a distance r from the center?

r>R is that from the center of the sphere?

 

[joemama69]

the exact wording is

Find the electric potential difference between the center of the sphere and a distance r away fr the center for both r<R and r>R. Be sure t indicate where you have chsen yur zero reference potential.

the example in the book finds the potential for r>R and they integrate from r_a to r_b, setting r_b = infinity

there is not really an exaple for r<R so I am lost

is there a rule for doing this, the book is not clear

 

[zachzach]

the exact wording is

Find the electric potential difference between the center of the sphere and a distance r away fr the center for both r<R and r>R. Be sure t indicate where you have chsen yur zero reference potential.

the example in the book finds the potential for r>R and they integrate from r_a to r_b, setting r_b = infinity

there is not really an exaple for r<R so I am lost

is there a rule for doing this, the book is not clear

Well if you are setting your zero potential to be at r = infinity then for r>R you would do the same integration. For r<R you still have to integrate from r = infinity but to a point r inside the sphere. This requires two different integrals because there are two different E fields (one for r>R and one for r<R).