Charge Densities & Dirac's Delta Function

[jdwood983]

Homework Statement

What is the (volume) charge density of a ring of radius $$r_0$$ and uniform charge density $$\lambda$$?

Homework Equations

The Dirac Delta Function

The Attempt at a Solution

I've done a few line charge densities of straight wires along an axis (usually z, but on x as well), but I'm getting stuck at using the delta functions when wrapping the wire around in a circle. I am pretty sure I'll want the ring to be lying in the y-z plane, as the problem continues with an integral with $$\cos\theta$$ in the integrand and $$\theta$$, being measured from the z-axis, should give me a delta function there to make the integral easier.

Any suggestions?

 

[gabbagabbahey]

Unless you are told otherwise, you are free to choose whatever coordinate system you like. Personally, I'd use cylindrical coordinates $\{r,\theta,z\}$ oriented so that the ring is in the xy-plane and centered on the origin...when you do this, the ring has zero extent in both the $z$ and radial directions, so you would expect the volume charge density to be of the form $\rho(\textbf{x})\propto\delta(r-r_0)\delta(z)$...I'll leave it up to you to find the constant of proportionality by means of a suitable integration...

 

[jdwood983]

I don't think that using cylindrical coordinates will work for me in this case as the "problem continues" part involves the ring existing between two grounded spheres of differing radii, which requires spherical coordinates.Small note, which really is a bit of nit-picking from a newbie poster to a certified Homework Helper, but you should use $$(\rho,\phi,z)$$ for cylindrical coordinates and not $$(r,\theta,z)$$ due to the similarity to spherical coordinates and the confusion it can bring using your method.

Also, the constant of proportionality, in cylindrical coordinates, would be, $$\frac{\lambda}{2\pi r}$$ ;)

 

[gabbagabbahey]

I don't think that using cylindrical coordinates will work for me in this case as the "problem continues" part involves the ring existing between two grounded spheres of differing radii, which requires spherical coordinates.

If you know the volume charge density in cylindrical coordinates, what's to stop you from transforming it to the appropriate spherical coordinate system?

Small note, which really is a bit of nit-picking from a newbie poster to a certified Homework Helper, but you should use $$(\rho,\phi,z)$$ for cylindrical coordinates and not $$(r,\theta,z)$$ due to the similarity to spherical coordinates and the confusion it can bring using your method.

There are only so many letters in the Greek and Latin alphabets. Using $\rho$ as the radial coordinate can also be confusing, since it is the same letter typically used for the volume charge density. There is no harm in using $\{r,\theta,z}$ as long as it is made clear that $r$ in this context, is the distance from the z-axis. Different authors use different notations with varying degrees of sloppiness, so a student must always look to the context in which a variable is used, to understand what it represents.

Also, the constant of proportionality, in cylindrical coordinates, would be, $$\frac{\lambda}{2\pi r}$$ ;)

Are you sure about that?

 

[jdwood983]

If you know the volume charge density in cylindrical coordinates, what's to stop you from transforming it to the appropriate spherical coordinate system?

Good point, working on that now

Are you sure about that?

See, this goes back to the spherical/cylindrical units--there shouldn't be the r in the denominator as lambda has units of charge/meter and each delta function has units of 1/meter making the three combined to be charge/meter^3; integrating this over all space gives charge, as it should. whoops!

 

[gabbagabbahey]

See, this goes back to the spherical/cylindrical units--there shouldn't be the r in the denominator as lambda has units of charge/meter and each delta function has units of 1/meter making the three combined to be charge/meter^3; integrating this over all space gives charge, as it should. whoops!

The $2\pi$ also isn't necessary... Integrating the linear charge density over the ring should give the total charge, as should integrating the volume charge density over all space...perform the integrations and compare the results.

 

[jdwood983]

You're right, did the integral and got the constant=$$\lambda$$.

Back to the original problem, though I didn't convert from cylindrical to spherical because it didn't look quite right, I end up with a charge density of

$$\rho(\mathbf{r})=\frac{\lambda\pi}{2r_0}\delta(r-r_0)\left[\delta\left(\phi-\frac{3\pi}{2}\right)+\delta\left(\phi-\frac{\pi}{2}\right)\right]$$

where $$\theta$$ is the angle sweeping from +z to -z and $$\phi$$ sweeps from +x towards +y. This still seems off to me, but $$\theta$$ runs from $$0\rightarrow\pi$$ so a point sticking out at radius $$r_0$$ and sweeping down $$\theta$$ at $$\phi=\pi/2$$ and $$\phi=3\pi/2$$ should give a circle on the y-z plane, right?

The constant of proportionality then comes from

$$Q=2\pi r_0\lambda=M\int_0^\infty r^2dr\delta(r-r_0)\int_0^{2\pi}d\phi\left[\delta\left(\phi-\frac{3\pi}{2}\right)+\delta\left(\phi-\frac{\pi}{2}\right)\right]\int_{-1}^1d(\cos\theta)$$

giving

$$M=\frac{2\pi r_0 \lambda}{r_0^24}=\frac{\lambda\pi}{2r_0}$$

Does this make sense to you, because it still seems a little funny to me.

 

[gabbagabbahey]

You should double check your value of $M$, but the general form of $\rho$ looks fine to me.