Can someone explain to me why E is constant?

[Demon117]

If an electrostatic field is parallel to the z-direction everywhere in some volume why is it that the field is constant without a charge in the volume?

Is this because \nabla\cdot E = 0 without the charge in the volume?

 

[mfb]

Right. If E is parallel to z everywhere, any deviation from a constant field strength would violate div E = 0.

 

[Darwin123]

If an electrostatic field is parallel to the z-direction everywhere in some volume why is it that the field is constant without a charge in the volume?

Is this because \nabla\cdot E = 0 without the charge in the volume?

An electrostatic field is constant in time regardless of the direction it is pointing or the charges in the volume. The word static means constant. Electrostatic by definition is the condition of having an electric field that is constant in time.
Maybe you asking why it is constant in space as well as time. I think that you mean that the electric field can't vary with z.
You mentioned the differential equation:
\nabla\cdot E = 0
I solved the equation for an electric field with only a z component, not an x or y component. I used a factor method for solving the differential equation. I don't have the text to write equations. I found that the solution did not vary with z. However, there is no constraint on how the electric field varies in x or y.
Although the electrostatic field in the z direction is constant in time, and constant in z, it does not have to be constant in x or y. The x and y variation will depend on the boundary conditions.

 

[mfb]

@Darwin123: He specified that the E-field is parallel to z, therefore the x- and y-components have to be 0 everywhere.

Therefore, \vec{E}=(0,0,E_z)^T and 0 = \nabla E = \partial_z E_z

In addition, he specified "electrostatic", therefore we know 0 = \mathrm{rot} \vec{E} = (\partial_y E_z, -\partial E_z,0)^T, leading to \partial_x E_z = \partial_y E_z = \partial_z E_z = 0. This is equivalent to a field which is constant in space everywhere.

 

[Darwin123]

@Darwin123: He specified that the E-field is parallel to z, therefore the x- and y-components have to be 0 everywhere.

Therefore, \vec{E}=(0,0,E_z)^T and 0 = \nabla E = \partial_z E_z

In addition, he specified "electrostatic", therefore we know 0 = \mathrm{rot} \vec{E} = (\partial_y E_z, -\partial E_z,0)^T, leading to \partial_x E_z = \partial_y E_z = \partial_z E_z = 0. This is equivalent to a field which is constant in space everywhere.

1) Are you sure that the curl of the electric field is a zero vector for an electrostatic system?
I was including the possibility that of the electric field being an electromotive force induced by a changing magnetic field. The word electrostatic may have thrown me off.
I thought the word electrostatic only referred to an electric field that was constant in time. The electric field could be induced by a magnetic field that was changing in time.

Although the electric field would be static, the magnetic field would not be static. So the system would not be magnetostatic, but it would be electrostatic. With a changing magnetic field, the curl of the electric field would not be zero. However, the electric field would still not vary with z.

This may be a question of semantics, but I am still interested.
2) Does the word electrostatic simply mean a constant electric field, or does it automatically include a constant magnetic field?
3) Does the word magnetostatic simply mean a constant magnetic field, or does it automatically include a constant electric field?
4) Would the phrase "electromagnetically static" have any value, or is it redundant?
5) Isn't it amazing one can still trip over words after years of using them?

 

[mfb]

According to wikipedia and basically every physics book:

Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving (without acceleration) electric charges. It is the branch of physics which deals with the study of charges at rest.

Without accelerating charges, you cannot have a magnetic field, and therefore rot E = 0.

 

[andrien]

maxwell eqn says that curl of electric field is zero ,if time varying magnetic fields are not present.

 

[Darwin123]

According to wikipedia and basically every physics book:

Without accelerating charges, you cannot have a magnetic field, and therefore rot E = 0.

Without moving charges, you can not have a magnetic field. Acceleration isn't necessary. Charges that move at a constant velocity can generate a magnetic field. As regards the OP's question, it doesn't matter.
The OP asked about the electric field in a finite volume that has no charges. However, have to be charges outside this volume to generate the electric field. If any of these charges outside the volume are moving, then there could be a magnetic field inside the volume.
Also, he didn't say whether this region is open or closed. If it is open, the boundary is not part of the volume. There could be charges right on the boundary.
In any case, the physics part of the question has been answered. If in a certain volume of empty space there is a constant electric field pointing in the z direction, no electric charges and no magnetic field, then the electric field is constant in time and space within this volume. The reason is that both the divergence and the curl of the electric field under these conditions are zero.