Finding normalization constant for proton charge distributio

[ElectricEel1]

Homework Statement

For a model of a proton's charge distribution,

:
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  I have to find the constant of normalisation for rho.

Homework Equations

The Attempt at a Solution

[/B]
I wrote p(r)=p_0 * (e^-r/R)/r

I then wrote it as
p_0^2 * integral from -infinity to +infinity of (e^-(r/)^2)/r^2 = 1

i tried to change variable by making z = r/R but then i was left with

1/R*p_0^2* integral of e^(-z^2) * (z^-2) = 1

I know that the integral of e^(-z^2) is the square root of pi but I don't know how to evaluate this with the z^-2 as well.

Have I started this correctly at all?

Thanks

 

[blue_leaf77]

You made mistake when calculating ##(e^{-r/R})^2##.

 

[TSny]

p_0^2 * integral from -infinity to +infinity of (e^-(r/)^2)/r^2 = 1

What does this integral represent? Why are you squaring the charge density? Why should the integral equal 1?

 

[ElectricEel1]

What does this integral represent? Why are you squaring the charge density? Why should the integral equal 1?

Part of my problem I think is that I'm not 100% sure on why I'm doing it. In my quantum work before the wave function was squared and multiplied by its complex conjugate and summed over infinity because the probability of finding it there will be 1. (I think). So I've just tried to use the same kind of method for this problem. Thanks

 

[blue_leaf77]

What does this integral represent? Why are you squaring the charge density? Why should the integral equal 1?

You are right, I completely misunderstood that radial function to be a wavefunction, didn't pay attention that it's a charge distribution already.

 

[ElectricEel1]

So should I not have squared this at all?

 

[blue_leaf77]

What have you learned from your electrodynamics course about charge distribution? For example given total charge ##Q##, its density ##\rho(\mathbf r)##, and volume ##V##, what's the equation connecting these quantities?

 

[kuruman]

Your attempt at a solution is incomplete. What did you use for dV in your integral?

 

[ElectricEel1]

What have you learned from your electrodynamics course about charge distribution? For example given total charge ##Q##, its density ##\rho(\mathbf r)##, and volume ##V##, what's the equation connecting these quantities?

oh yeah

rho = total charge / volume right or (ze)/(4/3pi*R^3) ?

and rho in this problem also = constant * (e^(r/R))/r

I think I need a volume integral here somewhere but I've not quite reached setting it up yet

 

[kuruman]

rho = total charge / volume right or (ze)/(4/3pi*R^3) ?

This is true only if the volume charge density is constant, i.e. independent of r. Here it is not because it is given to you as a function of r.

I think I need a volume integral here somewhere but I've not quite reached setting it up yet

You do need a volume integral that looks like ## \int \rho (r) dV ##. You know what ## \rho (r) ## looks like. What about ## dV ##?

 

[ElectricEel1]

or maybe that means that i should integrate

This is true only if the volume charge density is constant, i.e. independent of r. Here it is not because it is given to you as a function of r.

You do need a volume integral that looks like ## \int \rho (r) dV ##. You know what ## \rho (r) ## looks like. What about ## dV ##?

dV would look like r^2 sin theta right? if i do this then doesn't that mean that the integral is equal to the total charge?

 

[kuruman]

dV would look like r^2 sin theta right?

Not right. A volume element dV in spherical coordinates is more than just r²sinθ. What is the correct form?

if i do this then doesn't that mean that the integral is equal to the total charge?

Only if you integrate over the correct limits of integration. What are these limits?

 

[ElectricEel1]

Not right. A volume element dV in spherical coordinates is more than just r²sinθ. What is the correct form?

Only if you integrate over the correct limits of integration. What are these limits?

r^2 sin theta dr dtheta dphi

limits for r theta and phi are 0 to infinity, 0 to pi, and 0 to 2pi, respectively. but why do I need to find total charge to find this constant of normalisation?

thanks for your patience and advice

 

[kuruman]

but why do I need to find total charge to find this constant of normalisation?

The proton has charge e = 1.6×10^-19 C. The integral ## \int \rho (r) dV ## with the charge density you have will not give you that value or units. You need to multiply the charge distribution by normalization constant N so that the integral indeed gives you that value. This is what normalization means. When you have your answer, you should do a dimensional analysis to verify that your normalized ## \rho (r) ## has dimensions of charge/volume.

I am not sure that the upper integration limit for r extends all the way to infinity. What is the model for the proton that you are using according to the statement of the problem? Is the proton modeled as extending all the way out to infinity?

 

[ElectricEel1]

The proton has charge e = 1.6×10^-19 C. The integral ## \int \rho (r) dV ## with the charge density you have will not give you that value or units. You need to multiply the charge distribution by normalization constant N so that the integral indeed gives you that value. This is what normalization means. When you have your answer, you should do a dimensional analysis to verify that your normalized ## \rho (r) ## has dimensions of charge/volume.

I am not sure that the upper integration limit for r extends all the way to infinity. What is the model for the proton that you are using according to the statement of the problem? Is the proton modeled as extending all the way out to infinity?

at the bottom of the sheet it has a note saying that the charge distribution falls away as r approaches infinity and that this form is not very realistic but was chosen to simplify the maths so i assumed that meant that the limits were 0 and infinity. does that sound possible?

 

[kuruman]

so i assumed that meant that the limits were 0 and infinity. does that sound possible?

Yes, that's what it meant. Do you understand how to proceed from here?

 

[ElectricEel1]

Yes, that's what it meant. Do you understand how to proceed from here?

from there had p(r)=N*(e^-r/R)/r so had the integral of
N*(e^-r/R)/r * r^2 sin theta dr dtheta dphi = 1 and evaluated with limits 0>infinity,0>pi,0>2pi to get N = 1/(4*pi*R^2)

 

[kuruman]

Why is the integral equal to 1? You are not normalizing a probability which is equal to 1 when you add all probabilities over all space. You want to say that when you add up all the little charge elements ρdV over all space you get the total charge of the proton. So what should the other side of the equation be?

 

[ElectricEel1]

Why is the integral equal to 1? You are not normalizing a probability which is equal to 1 when you add all probabilities over all space. You want to say that when you add up all the little charge elements ρdV over all space you get the total charge of the proton. So what should the other side of the equation be?

so is it just equal to z*e or Q? the total charge?

 

[kuruman]

What's z for a proton?

 

[ElectricEel1]

What's z for a proton?

1!

 

[kuruman]

so is it just equal to z*e or Q? the total charge?

Correct. So, what is the RHS of the equation when you substitute 1 for z?

 

[ElectricEel1]

so right hand side is just e, or since the next question has 'q' in it i should probably use q

 

[kuruman]

Yes and yes.