# Electromagnetics - Surface Charge Density

• BigD959919
In summary, the conversation discusses solving for the electric field at the surface of a sphere using boundary conditions and Gauss's law. The potential and field at the surface are found, and the charge distribution is determined to be a dipole.
BigD959919
I answered my own question but thank you for the help.

Last edited:
I can't read your equations because you're not using LaTeX, but the general way to approach this would be to solve for the electric field at the surface of the sphere (only the normal component is non-vanishing, because it is a conductor) by first writing down boundary conditions on the potential, and then solving for the potential and finding the field from this.

Since the sphere is a conductor, it's surface forms and equipotential, so let the radius of the sphere be R, then you have two boundary conditions:

$\Phi(R) = \Phi_{0}$ where $\Phi_{0}$ is some constant which must be specified in the problem, if the sphere is grounded then it is zero. The other B.C. is:

$\Phi(\infinity) = E_{0}r\cos(\theta)$ this comes from the fact that at infinity the field behaves as if the sphere isn't there.

This is enough to get the potential, and the field at the surface. Once you have these things, then use gauss's law:

$$\nabla*E = \frac{\rho}{\epsilon_{0}}$$

to find the charge distribution.

(hint: You'll find that the sphere produces a dipole)

## 1. What is surface charge density in electromagnetics?

Surface charge density in electromagnetics is a measure of the amount of electric charge 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/m2).

## 2. How is surface charge density related to electric fields?

Surface charge density is directly related to the strength of an electric field at the surface of a material. It is proportional to the electric field strength and is given by the equation σ = ε0E, where ε0 is the permittivity of free space and E is the electric field strength.

## 3. What factors affect surface charge density?

The main factors that affect surface charge density include the material properties, such as its permittivity and conductivity, and the presence of any external electric fields or charges. The surface geometry and temperature of the material can also have an impact on surface charge density.

## 4. How is surface charge density measured?

Surface charge density can be measured using various techniques, including capacitance measurements, Kelvin probe microscopy, and surface potential measurements. These techniques involve using specialized equipment to measure the electric field or potential at the surface of a material, which can then be used to calculate the surface charge density.

## 5. Why is surface charge density important in electromagnetics?

Surface charge density plays a crucial role in many electromagnetics phenomena, such as the behavior of capacitors, the formation of electric double layers, and the interaction between materials and external electric fields. It is also an important concept in understanding the behavior of charged particles at surfaces and interfaces.

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