Electric Flux and Electric Flux Density

[CoolDude420]

Homework Statement

Hi,

So I'm doing a electromagnetics course and we've been given equations for electric flux and electric flux density but I can't seem to find any sort of intuitive explanation for these.

In my lecture notes, the electric flux density is introduced first as vector D and given the formula:

Vector D = epsilon*Vector E

The electric flux is defined as:

From my understanding from high school, the electric flux is the number of electric field lines passing through an area(perpendicular.)

I'm just really confused as to what is what in sort of a realistic viewpoint.

Homework Equations

The Attempt at a Solution

 

[haruspex]

the electric flux is the number of electric field lines

Yes, but field lines are just constructs to aid intuition. There are not actual discrete lines that can be counted. You can think of each field line as representing the same quantity of flux, but what that quantity is is up to you.

 

[kuruman]

D is proportional to E under most circumstances that you will encounter so if you understand one kind of flux, you should be able to understand the other. I suspect your question has more to do with Gauss's Law than with flux so I will discuss electric flux.

Imagine a closed surface like the skin of a potato. Now draw a square grid on the skin of the potato subdividing its area into many many little pieces dA. You can make the pieces as small as you like - we are doing calculus here. Number the pieces so that you can tell them apart. Go to piece 1 and measure the electric field at the location of that piece assuming that it is the same over the entire area of the piece. Consider a unit vector ##\hat{n}## perpendicular to the area pointing outwards, away from the "meat" of the potato. Find the component of the E-field, i.e. ##\vec{E_1} \cdot \hat{n_1}## and multiply by the area element ##dA_1##. Now go to element 2 and do the same. Add the new product ##\vec{E_2} \cdot \hat{n_2}~dA_2## to the previous one. Keep on adding until you run out of area elements. The sum of all the products is the electric flux.

OK, but what does that mean intutively? Remember that the dot product between two vectors is positive if the angle between the vectors is less than 90^o and negative if the angle is greater than 90^o. So, if the sum you get is positive, this means that more field lines on average are coming out of the area than going in; this is means that there is a source of field lines inside the meat of the potato. If the sum is negative, more field lines on average are going into the area than are coming out; this means that there is a sink of field lines inside the meat of the potato. And If the sum is zero, this means that there is neither a source nor a sink of electric field lines inside the meat of the potato.

Gauss's' Law asserts that the sum you get this way is proportional to the total net charge inside the meat of the potato. In other words, just by walking around the skin of the potato, keeping track of what goes in and what comes out, you can figure out what's under the skin without looking.

 

[CoolDude420]

D is proportional to E under most circumstances that you will encounter so if you understand one kind of flux, you should be able to understand the other. I suspect your question has more to do with Gauss's Law than with flux so I will discuss electric flux.

Imagine a closed surface like the skin of a potato. Now draw a square grid on the skin of the potato subdividing its area into many many little pieces dA. You can make the pieces as small as you like - we are doing calculus here. Number the pieces so that you can tell them apart. Go to piece 1 and measure the electric field at the location of that piece assuming that it is the same over the entire area of the piece. Consider a unit vector ##\hat{n}## perpendicular to the area pointing outwards, away from the "meat" of the potato. Find the component of the E-field, i.e. ##\vec{E_1} \cdot \hat{n_1}## and multiply by the area element ##dA_1##. Now go to element 2 and do the same. Add the new product ##\vec{E_2} \cdot \hat{n_2}~dA_2## to the previous one. Keep on adding until you run out of area elements. The sum of all the products is the electric flux.

OK, but what does that mean intutively? Remember that the dot product between two vectors is positive if the angle between the vectors is less than 90^o and negative if the angle is greater than 90^o. So, if the sum you get is positive, this means that more field lines on average are coming out of the area than going in; this is means that there is a source of field lines inside the meat of the potato. If the sum is negative, more field lines on average are going into the area than are coming out; this means that there is a sink of field lines inside the meat of the potato. And If the sum is zero, this means that there is neither a source nor a sink of electric field lines inside the meat of the potato.

Gauss's' Law asserts that the sum you get this way is proportional to the total net charge inside the meat of the potato. In other words, just by walking around the skin of the potato, keeping track of what goes in and what comes out, you can figure out what's under the skin without looking.

Very nice explanation! I think I understand electric flux. But I'm still not too sure about electric flux density and why it is used as vector D everywhere instead of the actual electric flux? Also in my lecture notes, electric flux is defined as the the flux of the electric flux density D instead of the of being the electric flux of the electric field strength E

 

[kuruman]

It doesn't matter what kind of vector field you have. Flux is a mathematical construct. You can go through the procedure that I described, for any vector field, whether it is E, D, the magnetic field B, the velocity vector field v (in a river) or whatever.