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I'm trying to understand the relationship between electrostatic energy and the electric field so that I can compute electric field components from electrostatic energy components.

Is it correct to assume that $E_x=\frac {F_x}q $ where $ F_x$ is the force computed from the electrostatic energy?

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    $\begingroup$ It looks like $F_x$ is just the electrostatic force. I don't know the concept of "electrostatic energy component". $\endgroup$ – my2cts Aug 15 '19 at 19:49
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I'm trying to understand the relationship between electrostatic energy and the electric field so that I can compute electric field components from electrostatic energy components.

The energy associated with an electrostatic field is called electrostatic potential energy. All forms of energy, including electrostatic potential energy, are scalar quantities and do not have directional components. The electric field is a vector quantity and therefore has directional components. So you cannot compute electric field components from electrostatic potential energy components as the latter do not exist.

Is it correct to assume that $𝐸_{𝑥}=\frac{𝐹_𝑥}{𝑞}$ where $𝐹_𝑥$ is the force computed from the electrostatic energy?

No.

$E_x$ is the electric field in the $x$ direction and is a vector quantity. Its direction is by convention the direction of the force that a positive charge would experience if placed in the field. It is not electrostatic potential energy. $F_x$ is the electrostatic force in the x direction and is also a vector quantity. It, too, is not electrostatic potential energy.

Relationship between electrostatic potential energy and the electric field

Now, to help understand this relationship, consider the following using the analogy of gravitational potential energy:

When charge is moved in an electric field its electrostatic potential energy either increases or decreases. This is analogous to moving a mass in a gravitational field which results in an increase or decrease in gravitational potential energy.

The work involved in moving a charge $q$ a distance $x$ given a constant electric field $E_x$, is given by

$$W=F_{x}x=qE_xx$$

The gravity analogy is

$$W=mgh$$

The sign for the work in each case will depend on whether the force acts in the same or opposite direction to the displacement of the charge or mass.

A force applied by an external agent does positive work to move charge in a direction opposite to the direction of the force exerted by the electric field on the charge, such as moving positive charges towards each other. This is analogous to the work done by an external force to lift an object in opposition to the downward force of gravity. At the same time, the electric field does an equal amount of negative work on the charge taking the energy the external agent gave the charge and storing it as electrostatic potential energy. This is analogous to the gravitational field doing negative work on the object being lifted taking the energy provided by the external force and storing it as gravitational potential energy.

On the other hand, if the work is done by the electric field on a charge placed in the field, the direction of the electrostatic force is the same as the movement of the charge, and the force gives the charge kinetic energy at the expense of electrostatic potential energy. This is analogous to the gravitational field doing positive work on a falling object giving it kinetic energy at the expense of gravitational potential energy.

Hope this helps.

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  • $\begingroup$ Thanks - that cleared everything up for me! $\endgroup$ – anon33 Aug 15 '19 at 23:26
  • $\begingroup$ @anon33 you are very welcome $\endgroup$ – Bob D Aug 15 '19 at 23:35
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In that equation $F_x$ is electrostatic force, not electrostatic energy. Furthermore, energy is not a vector so it does not have components. Force is a vector so it does have components.

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  • $\begingroup$ Okay so from the electrostatic energy one can compute the force, and from the force the corresponding components, and finally the Efield components from the relationship written above? $\endgroup$ – anon33 Aug 15 '19 at 19:53
  • $\begingroup$ No. The field is a different vector at every point in space. The energy stored in that entire field is a single number. You can’t compute an infinite number of vectors from one number! Perhaps you are confusing electrostatic energy with electrostatic potential. You can compute the field from the potential. $\endgroup$ – G. Smith Aug 15 '19 at 19:57

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