Induced surface charge density

• Silviu
In summary, in the given problem, the surface charge density induced by a point source above an infinite conducting plane with 0 potential can be calculated using the method of image charges. The electric field is 0 in the -z region because the potential of the plane is 0 and it is connected to Earth, resulting in no E field between them. This leads to no charges on the bottom of the plane and the necessary charges on the top are provided by the Earth.
Silviu

Homework Statement

Calculate the surface charge density induced by a point source above an infinite conducting plane, with 0 potential.

Homework Equations

##E=-\nabla V##
##V=\frac{q}{4\pi \epsilon_0 r}##

The Attempt at a Solution

I used the method of image charges and I calculated the potential induced by the charge and its mirror image ##V(x,y,z)##. Then I took a box around the plane and I applied the Gauss law. However, In their solution, assuming that the charge is in the ##+z## region and the surface is in the xy plane, they say that the E is 0 in the ##-z## area. From here ##\sigma = \epsilon_0 E = -\epsilon_0 \partial_z V##. I got the same result, but with a factor of 2 difference. My question is, why is the electric field 0 in the ##-z## region?
Thank you!

Plate is grounded. There will be a surface charge distribution topside only. The solution you found is only valid for z > 0 .

Silviu said:
My question is, why is the electric field 0 in the ##-z## region?
Thank you!
If the potential of the plane is zero then you can think of the plane as connected to Earth by a wire. The potential of Earth is zero. So there can be no E field between Earth and the plane. No E field, no E flux lines from Earth to the plane, ergo no charges on the plane's bottom to terminate those lines.

(The necessary charges on the top of the plane are provided by the Earth.)

What is induced surface charge density?

Induced surface charge density refers to the accumulation of electric charge on the surface of a material due to an external electric field. It is a result of the movement of free electrons within the material in response to the electric field.

What factors affect the magnitude of induced surface charge density?

The magnitude of induced surface charge density is affected by the strength of the external electric field, the distance between the charged object and the surface, and the properties of the material such as its conductivity and dielectric constant.

How is induced surface charge density different from fixed surface charge density?

Fixed surface charge density refers to the charge that is permanently present on the surface of a material, while induced surface charge density is temporary and only present when an external electric field is applied.

What are the applications of induced surface charge density?

Induced surface charge density has many practical applications, including the operation of capacitors, the functioning of touchscreens, and the separation of particles in electrophoresis.

How is induced surface charge density related to the concept of electric potential?

Induced surface charge density is directly related to the electric potential of a material. When an external electric field is applied, it creates a potential difference between the surface of the material and its interior, which leads to the accumulation of induced surface charge density.

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