How Can Surface Charge Density Equal bcos(θ)?

In summary: So in summary, the surface charge density is defined by a hemisphere of radius b on the x-y plane, and is given by \rho_s(z) = \rho_0z or \rho_s(z)\rightarrow\rho_0cos(\theta), where \rho_0 is a constant with units of C/m^2. The variable b has units of meters and is used to calculate the charge density at points with coordinates (x,y,z).
  • #1
jeff1evesque
312
0

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 [tex]\rho_s(z) = z (\frac{Coul}{m^3})[/tex].

Homework Equations


[tex]\rho_s(z) = z = Rcos(\theta) = bcos(\theta) (\frac{Coul}{m^3})[/tex].


3. Question
My question is how can the surface charge density equal to [tex]Rcos(\theta) = bcos(\theta)[/tex]? 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 [tex]\neq (\frac{Coul}{m^3})[/tex].

Thanks,


JL
 
Physics news on Phys.org
  • #2
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.)
 
  • #3
dx said:
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.
 
  • #4
dx said:
... 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.

jeff1evesque said:
[tex]\rho_s(z) = z = Rcos(\theta) = bcos(\theta) (\frac{Coul}{m^3})[/tex].
This is misleading (i.e. wrong). The first inequality should be confusing. I would suggest:
[tex]
\rho_s(z)=\rho_0z\rightarrow{}\rho_0Rcos(\theta)=bcos(\theta) (C/m^3)
[/tex]
where [itex]\rho_0[/itex] is some unknown constant that has units of charge-per-volume and R and b have units of length.
Or, better yet,
[tex]
\rho_s(z)\rightarrow\rho_0cos(\theta)
[/tex]
where [itex]\rho_0[/itex] is some unknown constant with units of C/m^2.
 
  • #5
turin said:
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:
[tex]
\rho_s(z)=\rho_0z\rightarrow{}\rho_0Rcos(\theta)=bcos(\theta) (C/m^3)
[/tex]
where [itex]\rho_0[/itex] is some unknown constant that has units of charge-per-volume and R and b have units of length.
Or, better yet,
[tex]
\rho_s(z)\rightarrow\rho_0cos(\theta)
[/tex]
where [itex]\rho_0[/itex] is some unknown constant with units of C/m^2.

That makes much more sense, thank you.
 

FAQ: How Can Surface Charge Density Equal bcos(θ)?

1. What is surface charge density?

Surface charge density is a measure of the electric charge present per unit area on the surface of a material. It is typically denoted by the symbol σ and is expressed in units of coulombs per square meter (C/m^2).

2. How is surface charge density calculated?

Surface charge density can be calculated by dividing the total electric charge on the surface by the total surface area. It can also be calculated by multiplying the electric field strength (E) by the permittivity of free space (ε0). The formula for surface charge density is σ = Q/A = ε0 * E.

3. What factors affect surface charge density?

The surface charge density of a material can be affected by various factors, such as the type and amount of charge present, the surface area of the material, and the dielectric constant of the surrounding medium. Additionally, the presence of other electric charges nearby can also affect the surface charge density.

4. Why is surface charge density important in electrostatics?

Surface charge density plays a crucial role in understanding and predicting the behavior of electric charges on the surface of a material. It helps in determining the electric field strength and potential at a given point, as well as the forces and interactions between charged objects. In many practical applications, such as in the design of electronic devices, the control and manipulation of surface charge density is essential.

5. How is surface charge density related to surface potential?

Surface charge density and surface potential are closely related. The surface potential is a measure of the electric potential at a particular point on the surface of a material, while the surface charge density is a measure of the electric charge present on the surface. In fact, the surface potential can be calculated by dividing the surface charge density by the permittivity of the surrounding medium.

Similar threads

Replies
1
Views
1K
Replies
6
Views
2K
Replies
15
Views
4K
Replies
9
Views
2K
Replies
2
Views
2K
Replies
8
Views
1K
Back
Top