Work done by pushing a proton into the sphere with non-uniform charge

[Mr_Pu]

Homework Statement

There is a sphere, with radius r and charge density ρ(r) = B/r. This sphere has a small hole drill through the middle. How much work is done by pushing the proton from the edge of a sphere to the center, through that hole.

Relevant Equations

e = ∫ρ dV
U = e/ 4πεr
W = eU

I have already calculated full charge inside the sphere: e = ∫ρ dV = 2πBr^2
And I know that electric potential on the edge of the sphere is: U = e/ 4πεr

The idea is that I calculate work by the change of electric potential energy, but to do that, I have to calculate electric potential energy in the middle as well. And then I also don't know how exactly a proton is affected by that.

 

[jbriggs444]

You understand, I expect, that the potential at the edge of the sphere is calculated as an integral of field strength on a path running from infinity to the edge of the sphere. Of course, that integral has been worked for you, it is printed in the physics textbook. It has become one of the standard tools in your bag.

The problem at hand has a field strength that does not take the form of an inverse square. The canned formula from your bag of tricks won't get you the rest of the way from the edge of the sphere to its center.

So build a new formula.

You have the tools you need for that -- you've already calculated the total charge contained in a sphere of radius r.

 

[Mr_Pu]

You understand, I expect, that the potential at the edge of the sphere is calculated as an integral of field strength on a path running from infinity to the edge of the sphere. Of course, that integral has been worked for you, it is printed in the physics textbook. It has become one of the standard tools in your bag.

The problem at hand has a field strength that does not take the form of an inverse square. The canned formula from your bag of tricks won't get you the rest of the way from the edge of the sphere to its center.

So build a new formula.

You have the tools you need for that -- you've already calculated the total charge contained in a sphere of radius r.

However, the thing that troubles me, is that in the middle the density of charge limits to infinity and I'm not sure how to build out a formula for that.

 

[jbriggs444]

However, the thing that troubles me, is that in the middle the density of charge limits to infinity and I'm not sure how to build out a formula for that.

It is good that you are concerned. But the things that enter into a formula for field strength are total enclosed charge and radius. Does the field strength diverge as the center is approached?

And even if it does, does the path integral diverge as one evaluates it with an inner limit closer and closer to the center?

 

[Mr_Pu]

It is good that you are concerned. But the things that enter into a formula for field strength are total enclosed charge and radius. Does the field strength diverge as the center is approached?

And even if it does, does the path integral diverge as one evaluates it with an inner limit closer and closer to the center?

I think it doesn't.

 

[jbriggs444]

I think it doesn't.

I agree. I think you'll find that things cancel agreeably.

 

[Mr_Pu]

I agree. I think you'll find that things cancel agreeably.

When I compute the integral for electric field inside a sphere ∫E dS = ∫ de / ε I get that E = 2B(r^2 - R^2)/r^2 which doesn't make any sense to me, since when R = 0, E is just 2B

 

[jbriggs444]

When I compute the integral for electric field inside a sphere ∫E dS = ∫ de / ε I get that E = 2B(r^2 - R^2)/r^2 which doesn't make any sense to me, since when R = 0, E is just 2B

Can you show your work? I do not get anything of the sort.

 

[Mr_Pu]

Can you show your work? I do not get anything of the sort.

E dS = de/ε --> E ∫2πrdr = ∫(B/r ⋅ 4πr^2 dr)/ε --> Eπr^2 = 2πB(r^2 - R^2) --> E = 2B(r^2 - R^2)/r^2

 

[jbriggs444]

E dS = de/ε --> E ∫2πrdr = ∫(B/r ⋅ 4πr^2 dr)/ε --> Eπr^2 = 2πB(r^2 - R^2) --> E = 2B(r^2 - R^2)/r^2

Can you justify that very first equality? You have a potential ($E$) multiplied by a displacement ($dS$) on the left. What does that denote? Do you not want a field strength times an incremental displacement? Or a potential divided by an incremental displacement?

Personally, I'd start by invoking Coulomb's law and deducing field strength from enclosed total charge and radius. Then I'd integrate field strength from zero to R to get the change in potential between those limits.

 

[Mr_Pu]

Can you justify that very first equality? You have a potential ($E$) multiplied by a displacement ($dS$) on the left. What does that denote? Do you not want a field strength times an incremental displacement? Or a potential divided by an incremental displacement?

Personally, I'd start by invoking Coulomb's law and deducing field strength from enclosed total charge and radius. Then I'd integrate field strength from zero to R to get the change in potential between those limits.

Can you please show your work, I'm kind of lost here.

 

[jbriggs444]

I don't understand.

Well, let's start with enclosed total charge in a sphere of radius r. You derived that in post #1.

Given the charge in that sphere then Newton's spherical shell theorem has something to say about the effect of that charge distribution. And Coulombs law has something to say about the force on a 1 unit test charge at radius r outside such a charge distribution.

Can you write a formula for that?

Perhaps we should clarify some symbols. The problem statement completely bollixed that part up. The complete spherical distribution has radius "$R$" (not $r$). We are considering the electrostatic force on a unit charge at radius "$r$" (not $R$) somewhere in the interior of that distribution.

 

[Mr_Pu]

Well, let's start with enclosed total charge in a sphere of radius r. You derived that in post #1.

Given the charge in that sphere then Newton's spherical shell theorem has something to say about the effect of that charge distribution. And Coulombs law has something to say about the force on a 1 unit test charge at radius r outside such a charge distribution.

Can you write a formula for that?

Perhaps we should clarify some symbols. The problem statement completely bollixed that part up. The complete spherical distribution has radius "$R$" (not $r$). We are considering the electrostatic force on a unit charge at radius "$r$" (not $R$) somewhere in the interior of that distribution.

Well, let's start with enclosed total charge in a sphere of radius r. You derived that in post #1.

Given the charge in that sphere then Newton's spherical shell theorem has something to say about the effect of that charge distribution. And Coulombs law has something to say about the force on a 1 unit test charge at radius r outside such a charge distribution.

Can you write a formula for that?

Perhaps we should clarify some symbols. The problem statement completely bollixed that part up. The complete spherical distribution has radius "$R$" (not $r$). We are considering the electrostatic force on a unit charge at radius "$r$" (not $R$) somewhere in the interior of that distribution.

So if I understand correctly, I should write the Coloumbs law, for the charge distributed on a sphere and then limit that to r=0?

 

[rude man]

Homework Statement::

I have already calculated full charge inside the sphere: e = ∫ρ dV = 2πBr^2
And I know that electric potential on the edge of the sphere is: U = e/ 4πεr

The idea is that I calculate work by the change of electric potential energy, but to do that, I have to calculate electric potential energy in the middle as well. And then I also don't know how exactly a proton is affected by that.

You are on the right track. You have already calculated U at the sphere's edge. Why not calculate it at its center? Then,use your formula for $\Delta W$ assuming you know the atomic number of the proton.