- #1

Sebas4

- 13

- 2

- Homework Statement
- Boundary conditions electrostatics.

- Relevant Equations
- Meaning of the electric field variables in the boundary condition equations.

Hey, I have a really short question about electrostatics.

The boundary conditions are :

[tex] \mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} [/tex],

[tex] \mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}[/tex].

My question is what is [itex] \mathbf{E}^{\perp }_{above} [/itex], [itex] \mathbf{E}^{\perp }_{below} [/itex],

[itex] \mathbf{E}^{\parallel }_{above}[/itex] and [itex]\mathbf{E}^{\parallel}_{below}[/itex], is it the

So is the electric field in this equation the sum of the external field and the electric field due to the charge at the boundary?

I will try to explain my question with an example.

Let's say we have an infinite plane with homogeneous charge density [itex]\sigma[/itex].

The electric field above the plane

[tex] \mathbf{E} = \frac{\sigma}{2\varepsilon_{0}}\mathbf{\hat{z}} [/tex].

The electric field below the plane is

[tex] \mathbf{E} = - \frac{\sigma}{2\varepsilon_{0}}\mathbf{\hat{z}} [/tex].

We have a homogeneous external field pointing in the z-direction, [itex]\mathbf{E}_{external} = \mathbf{E}_{0} \mathbf{\hat{z}}[/itex].

The electric field just below the surface of the plane is

[tex] \mathbf{E}_{total below} = \left(\mathbf{E}_{0} - \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} [/tex].

The electric field just above the surface of the plane is

[tex] \mathbf{E}_{total above} = \left(\mathbf{E}_{0} + \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} [/tex].

If we plug this in, in the boundary condition we get

[tex] \mathbf{E}_{total above} - \mathbf{E}_{total below} = \left(\mathbf{E}_{0} + \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} - \left(\mathbf{E}_{0} - \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} = \frac{\sigma}{\varepsilon_{0}} \mathbf{\hat{z}} [/tex].

This is true, according to the boundary condition.

I have also another question, this also works for non-homogeneous charge density boundaries? (I think so).

The boundary conditions are :

[tex] \mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} [/tex],

[tex] \mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}[/tex].

My question is what is [itex] \mathbf{E}^{\perp }_{above} [/itex], [itex] \mathbf{E}^{\perp }_{below} [/itex],

[itex] \mathbf{E}^{\parallel }_{above}[/itex] and [itex]\mathbf{E}^{\parallel}_{below}[/itex], is it the

**total**electric field component near the boundary?So is the electric field in this equation the sum of the external field and the electric field due to the charge at the boundary?

I will try to explain my question with an example.

Let's say we have an infinite plane with homogeneous charge density [itex]\sigma[/itex].

The electric field above the plane

[tex] \mathbf{E} = \frac{\sigma}{2\varepsilon_{0}}\mathbf{\hat{z}} [/tex].

The electric field below the plane is

[tex] \mathbf{E} = - \frac{\sigma}{2\varepsilon_{0}}\mathbf{\hat{z}} [/tex].

We have a homogeneous external field pointing in the z-direction, [itex]\mathbf{E}_{external} = \mathbf{E}_{0} \mathbf{\hat{z}}[/itex].

The electric field just below the surface of the plane is

[tex] \mathbf{E}_{total below} = \left(\mathbf{E}_{0} - \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} [/tex].

The electric field just above the surface of the plane is

[tex] \mathbf{E}_{total above} = \left(\mathbf{E}_{0} + \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} [/tex].

If we plug this in, in the boundary condition we get

[tex] \mathbf{E}_{total above} - \mathbf{E}_{total below} = \left(\mathbf{E}_{0} + \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} - \left(\mathbf{E}_{0} - \frac{\sigma}{2\varepsilon_{0}}\right) \mathbf{\hat{z}} = \frac{\sigma}{\varepsilon_{0}} \mathbf{\hat{z}} [/tex].

This is true, according to the boundary condition.

I have also another question, this also works for non-homogeneous charge density boundaries? (I think so).