Electric Charge Density (1 Viewer)

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

(src: Intro to Electrodynamics, Griffith, Problem 1.46a)
Q: Write an expression for the electric charge density [itex] \rho (r)[/itex] of a point charge [itex]q[/itex] at [itex]r^'[/itex]. Make sure that the volume integral of [itex]\rho[/itex] equals [itex]q[/itex].

Now, Closest I can seem to come up with is:


[tex] \rho(r)=\frac{q}{4*Pi*R^2}\delta(r-r^')[/tex]

But, the problem I see with this, is that while yes, integrating this over any volume [itex]V[/itex] that enclosed the point charge will return q, but that q would have to have units of charge/unit_volume which just dosent make sense. Or am I missing something?

Any help would be appreciated.
 

Tom Mattson

Staff Emeritus
Science Advisor
Gold Member
5,453
21
I think that delta function should be 3D: [itex]\delta^{(3)}(\vec{r}-\vec{r}')[/itex]. Note that the n-D delta function has dimensions of (length)-n.
 
Last edited:
Yeah, sorry missed that. Have the [itex]\delta^3[/itex] on my paper, just forgot to type it in.

I don't understand how n-D delta functions have a dimension of (length)-n, could you explain that perhaps?
 

Tom Mattson

Staff Emeritus
Science Advisor
Gold Member
5,453
21
Sure, let's look at the 1D case. Consider the following integral:

[tex]\int_{-\infty}^{\infty}\delta(x)dx=1[/tex]

The right side of that is 1. Not 1 meter or 1 Joule, just plain old 1. So if the units of dx are meters, then what must the units of the delta function be? Inverse meters.

Similar results hold for higher dimensional cases.
 
Okay, that makes sense.

Thanks for your help, this was driving me crazy, I couldnt figure out why units were not making sense.
 

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top