# Find charge density of simple electric field

doodlepin

## Homework Statement

Suppose an electric field E(x,y,z) has the form:

E_x = ax, E_y = 0, E_z = 0

where a is a constant. What is the charge density? How do you account for the fact that the field points in a particular direction, when the charge density is uniform?

## Homework Equations

charge density = charge per unit volume

gauss's law says the volume integral of the electric field around some closed surface is negative the charge enclosed divided by the epsilon constant

## The Attempt at a Solution

I picture this as an infinite charged plate going through x = 0 in the yz plane, which would have purely an electric field in the x-direction. So i think I am supposed to use gauss' law and integrate over all space? But i think that would get me some kind of infinite solution? Any hints would be greatly appreciated.

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Is this the complete statement of the problem? - word for word?

doodlepin
Yes. Except for a quick blurb in parenthesis saying: (This is a more subtle problem than it looks, and worthy of careful thought)

Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Suppose an electric field E(x,y,z) has the form:

E_x = ax, E_y = 0, E_z = 0

where a is a constant. What is the charge density? How do you account for the fact that the field points in a particular direction, when the charge density is uniform?

## Homework Equations

charge density = charge per unit volume

Gauss's law says the volume integral of the electric field around some closed surface is negative the charge enclosed divided by the epsilon constant

## The Attempt at a Solution

I picture this as an infinite charged plate going through x = 0 in the yz plane, which would have purely an electric field in the x-direction. So i think I am supposed to use gauss' law and integrate over all space? But i think that would get me some kind of infinite solution? Any hints would be greatly appreciated.
(My Gauss's Law has no negative in it. Electric field originates on positive charge and terminates on negative.)

Assuming that you have "an infinite charged plate going through x = 0 in the yz plane", then you don't need to integrate over the whole plate. Integrate over a cylinder having flat faces parallel to the yz plane, on face with positive x-coordinate, the other with negative x-coordinate. Other sides should be parallel to the x-axis (perpendicular to the yz plane).

If the area of a flat face is A, how much flux passes through each surface of the cylinder?

How much charge is enclosed by the cylinder?

How much area of the plate is enclosed by the cylinder?

doodlepin
(My Gauss's Law has no negative in it. Electric field originates on positive charge and terminates on negative.)

Assuming that you have "an infinite charged plate going through x = 0 in the yz plane", then you don't need to integrate over the whole plate. Integrate over a cylinder having flat faces parallel to the yz plane, on face with positive x-coordinate, the other with negative x-coordinate. Other sides should be parallel to the x-axis (perpendicular to the yz plane).

If the area of a flat face is A, how much flux passes through each surface of the cylinder?

How much charge is enclosed by the cylinder?

How much area of the plate is enclosed by the cylinder?

Flux that passes through each end of the cylinder:
E is independent of surface area at end of cylinder so flux is just E*A = ax*A
where A is the area of the end of the cylinder.

Charge enclose by the cylinder is charge density of infinite plane multiplied by A

Area of plate enclosed by the cylinder is A.

I don't have any further understanding of what the charge density of space is.

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