Deriving Electric Field Energy Density for Arbitrary Charge Distribution?

In summary, the conversation discusses deriving an expression for the electric field energy density of an arbitrary charge distribution using basic electrostatics. The equation for electric field energy density is provided as w = \frac{1}{2} \epsilon E^2, and the attempt at a solution involves computing the elementary work done by the field on an external electrical charge and dividing it by the elementary unit of volume in spherical coordinates. However, the solution has not been successful so far and the question remains on how to proceed.
  • #1
mystify
1
0

Homework Statement


Provide a proof for the electric field energy density of an arbitrary charge distribution using basic notions of electrostatics.


Homework Equations


The energy density for an electric field is [itex]w = \frac{1}{2} \epsilon E^2[/itex]


The Attempt at a Solution


I am not sure at all what kind of energy formula to derive. I computed the elementary work done by the field on an external electrical charge, then divided it by the elementary unit of volume in spherical coordinates, but to no avail.

How to proceed ?
 
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  • #2
mystify said:
I am not sure at all what kind of energy formula to derive.
I guess you are to derive an expression for the energy density at a point [itex]\vec{R} [/itex] due to a charge distribution given by [itex]\rho (\vec{r}) [/itex].
 

1. What is electric field energy density?

Electric field energy density is a measure of the amount of energy stored in an electric field per unit volume. It is a physical quantity that describes the amount of energy that can be transferred to a charged particle placed in the electric field.

2. How is electric field energy density calculated?

Electric field energy density is calculated by taking the square of the electric field strength and multiplying it by the electric constant (ε0) divided by 2. The formula is U = (ε0/2) * E², where U is the energy density and E is the electric field strength.

3. What is the unit of measurement for electric field energy density?

The unit of measurement for electric field energy density is joules per cubic meter (J/m³). This unit represents the amount of energy stored in an electric field per unit volume.

4. How does the electric field energy density affect the behavior of charged particles?

The electric field energy density affects the behavior of charged particles by determining the amount of potential energy they have in an electric field. This energy can be converted into kinetic energy, causing the particles to move. The direction and magnitude of the electric field also determine the direction and speed of the particle's motion.

5. How does the electric field energy density change with distance from a charged object?

The electric field energy density follows an inverse-square law, meaning that it decreases with the square of the distance from a charged object. This means that as the distance from the object increases, the energy density decreases, and vice versa. This relationship is important in understanding the behavior of electric fields and the effects they have on charged particles.

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