Need help understanding Gauss's Law

[1220Student]

Hi all,

I'm having some difficulty with electric flux.

Does electric flux stay the same if you double the diameter of a circle assuming the electric field is uniform?

Any help would be much appreciated.

Thanks

 

[Defennder]

Well, by Gauss law that depends on whether the enlarged circle (or sphere) encloses any charges. If it does not, then it is the same, otherwise it's different.

 

[1220Student]

if you don't have a charges inside the circle, if you double the area of the circle won't the electric flux also double (ie Electric flux = E.A)?

 

[Defennder]

What does Gauss law say about the flux through a closed surface? Why should it double?

EDIT: I assumed you confined the discussion to 2D, but I realized you may instead be talking about a circular plane and corresponding flux through that plane. Well if that is the case, note that you cannot use Gauss law here anymore since there is no closed surface. You now have to consider the direction of the electric field lines. In other words, you need to consider what the E-field lines look like and how the circular plane is positioned.

Assuming that the plane of the circle is not parallel to the direction of the field lines, then I would say that the flux through that circular plane is doubled when the area of the circular plane is doubled. But since your question was what happens when the diameter is doubled, then you should think of how much the area of the circle plane is increased when you double the diameter/radius.

 

[atyy]

if you don't have a charges inside the circle, if you double the area of the circle won't the electric flux also double (ie Electric flux = E.A)?

Let's suppose you are in a region in which is zero charge, but there is an electromagnetic wave whose wavefronts are perpendicular to its direction of propagation. Let's use Gauss's law with a closed surface that is a cube oriented with its walls parallel/perpendicular to the wave. By Gauss's law since no charge is enclosed, the flux through the cube is zero. But if we don't use Gauss's law, we can just look at the geometry of the wave and the cube. Basically, the wave enters the back of the cube and then exits the front, so the flux into the back is exactly subtracted by the flux out of the front, so the flux is zero. If you make the cube twice as big in area, then twice as much enters through the back, but also twice as much exits through the front, so the total flux is still zero.

Of course, this wouldn't work if there was no such thing as an electromagnetic wave whose wavefronts are perpendicular to the direction of propagation. But it appears that such things do exist (sunlight reaching the Earth is roughly a combination of many such waves), and luckily we know how to describe them by 5 equations (4 Maxwell equations + 1 Lorentz force law), one of which is Gauss's law.

How about more complicated fields? It turns out that as long as you are in any box shaped room that contains no charges, any possible field is some complicated combination of plane waves. As for rooms of other shapes, go ask an engineer.

 

[merryjman]

Does electric flux stay the same if you double the diameter of a circle assuming the electric field is uniform?

So you've got a circle sitting inside a uniform electric field. Let's keep it simple. It's important to know which direction the field points. Let's assume the circle "faces" the field, so that the most field lines go through the circle as possible. In that case, doubling the diameter will change the flux. How much? Well, how much does the area of the circle change? After all, that's what flux is: the "number" of field lines going through an area. So you are right that the flux changes, but you're wrong that the area doubles.

On the other hand, if the circle is rotated 90 degrees, no field lines go through it at all, so doubling its diameter does nothing.

 

[atyy]

So, for a uniform electric field, or plane electromagenetic wave, if you use a cube instead of a sphere, you can visualise that what goes in also comes out. For the sphere, you can visualise it as being made of many cubes of many different sizes.

How about if the sphere encloses a single charge, and there are no charges anywhere else? The flux is always equal to the charge within the sphere (it's true even if you have an arbitrary distribution of charges outside the sphere, but then it's harder to visualise why, because there is no spherical symmetry in the electric field within the sphere). If you increase the area of the sphere by increasing the radius r, the flux should stay the same. since the bigger sphere encloses the same amount of charge as the smaller sphere. How do we visualise this? The area is proportional to r^2, but by Coulomb's law, the electric field is inversely proportional to r^2, and compensates exactly for the increase in area. This wouldn't work if Coulomb's law contained the inverse of r^3, nor would Gauss's law be true then.