# Surface charge density

## Homework Statement

A surface is defined by a hemisphere of radius b, centered on the x-y plane. The surface charged density is given by $$\rho_s(z) = z (\frac{Coul}{m^3})$$.

## Homework Equations

$$\rho_s(z) = z = Rcos(\theta) = bcos(\theta) (\frac{Coul}{m^3})$$.

3. Question
My question is how can the surface charge density equal to $$Rcos(\theta) = bcos(\theta)$$? That is a measure of the distance from the origin to the surface [element], and thus only [to my knowledge] have units of radius b, or meters $$\neq (\frac{Coul}{m^3})$$.

Thanks,

JL

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dx
Homework Helper
Gold Member
z has no units, it's just a coordinate in the coordinate system, and what they are saying is that the charge density at points with coordinates (x,y,z) is equal to z C/m2. (You wrote C/m3 which I'm assuming is a typo since we're talking about a surface, not a 3D region.)

z has no units, it's just a coordinate in the coordinate system, and what they are saying is that the charge density at points with coordinates (x,y,z) is equal to z C/m2. (You wrote C/m3 which I'm assuming is a typo since we're talking about a surface, not a 3D region.)
Sorry about the typo. But what about the variable b which has units of meters? How does that fit into the interpretation of surface charges?

Thank you.

turin
Homework Helper
... the charge density at points with coordinates (x,y,z) is equal to z C/m2. (You wrote C/m3 which I'm assuming is a typo since we're talking about a surface, not a 3D region.)
No typo. One unit of length in z cancels one unit of length in the denom. of C/m^3.

$$\rho_s(z) = z = Rcos(\theta) = bcos(\theta) (\frac{Coul}{m^3})$$.
This is misleading (i.e. wrong). The first inequality should be confusing. I would suggest:
$$\rho_s(z)=\rho_0z\rightarrow{}\rho_0Rcos(\theta)=bcos(\theta) (C/m^3)$$
where $\rho_0$ is some unknown constant that has units of charge-per-volume and R and b have units of length.
Or, better yet,
$$\rho_s(z)\rightarrow\rho_0cos(\theta)$$
where $\rho_0$ is some unknown constant with units of C/m^2.

No typo. One unit of length in z cancels one unit of length in the denom. of C/m^3.

This is misleading (i.e. wrong). The first inequality should be confusing. I would suggest:
$$\rho_s(z)=\rho_0z\rightarrow{}\rho_0Rcos(\theta)=bcos(\theta) (C/m^3)$$
where $\rho_0$ is some unknown constant that has units of charge-per-volume and R and b have units of length.
Or, better yet,
$$\rho_s(z)\rightarrow\rho_0cos(\theta)$$
where $\rho_0$ is some unknown constant with units of C/m^2.
That makes much more sense, thank you.