- Carl Friedrich Gauss

Picture of the Man himself

There was an idea...by a german mathematician and physicist, The man who found out the a.p formula at age of 10 after his teacher tried to punish him for doing sums too fast by telling him to sum from 1 to 100 and also the discovered row elimination method in Matrices for solving systems of Linear equations, Carl Friedrich Gauss....To find the electric field given a certain charge distribution.





To truly understand this beautiful theorem, We need to understand field lines and look at charges from a different perspective.





1.)Vector Valued Functions & Field Lines!





Field lines are a thing which is so crucial and important to electrostatics but just thrown once at the students and then forgotten by everyone including the teacher. To properly understand them we need to learn about functions which relates each point in space to a vector, these functions are known as vector valued functions. If we graph the function for all points in the domain we get something known as a vector field, That is we input each point in space individually to the function and sketch the vectors associated each point on some coordinate system. The idea is the only important thing here, dont fret much about the details of how it would look like or how it would work.





p.s: The origin of each vector is at the input point,Say I input (1,1) into the function ,the vector which is outputed would be attached to that input point (1,1) (think of how the arms of a clock are hinged to the center)

Swirly vector field function graphed( fig.1)





We can observe that in the fig.1 that some vectors are longer than others, this means it has a greater numerical magnitude when you convert it from the algebraic form to magnitude form.





Now let us come to field lines. We can connect the vectors which are pointing in a near same direction to create a line( see fig.2)

Imagine a car running a long that vector path leaving it's trails!





A keen person would ask, don't we lose the aspect of magnitude if you use these curvey lines? Good question, Hah I said that to myself! Jokes aside, We increase the number of lines crossing a certain area if we want to show that the vectors which the line is passing through is of heavier magnitude. I shall illustrate this using some charge pairs. (this will make more sense as you read on)

The left most image shows a charge of (1,1) the middle most shows (2,-1) pair and the rightmost one shows a pair of (4,-4). Red is always positive and blue is negative. Do note eletric fields from dipole do not cancel.

When my physics teacher questioned the class what a charge is, I told him " It is an inherent property of matter which can be used to describe it's behavior" but for understanding this theorem we need to think of this in a different perspective. Let's remove that matter part and isolate the charge part and look at it separately,When I say charge, I want you to think of protons and electrons with their charges





It is a property well observed that if you have a charged particle it will automatically create a region of influence where it can effect the movement of other charged particles and be effected itself. This region of influence can be mathematically described by a quantity known as Electric field or E-field for short.





An Electric field is a vector quantity but using our field line idea we can also think of it as a number field lines bundled together, Remember that we draw more lines when the vector has greater magnitude. The more lines bundled together , the stronger the field.





The field line and vector duality might be weird and unpleasant for some but it is in fact a thing which actually exists in nature, try ponder over it if you can't accept it instantly. They are both equally correct. Now we will go on to speak about Electric flux.











Electric flux is the number of electric field lines passing perpendicular through a imaginary surface in free space, We will talk about real surfaces later. Or more formally we can describe flux as

Now for the new perspective on charges, Think of positive charges as sources(taps) of electric field lines and think of negative charges as sink which drains these field lines into itself. But why do positive charges release field lines and negative charges suck it into themselves? Ask that to mother nature and tell me what she says;Honestly even I don't know.

Sort of like this except the whole proton isn't sucked in by that one

sucy negative charge

3.) Pulling the field lines together, Gauss theorem!





Now finally we can understand the beauty of this jewel of an equation



Let us see what the left side says, I read 'The line integral of the electric field dot product area', We know the electric field dot area is just the flux and integrating it over the whole gaussian surface (an imaginary surface you make up,usually has the blessings of symmetry) in a line fashion, Imagine a line of area patches and we dot each one with electric field through that patch, this would give us the total flux. If that doesn't bode well with you just think that it we are summing up the the flux across the whole area of our surface.





The right side says the total charge inside the surface divided by a constant? How are these two things even equal?





Now comes in the magic! Collect your thoughts and remember the things discussed in the previous sections and brace yourselves! What gauss theorem is essentially telling us is that the amount of flow (fluxy flux) proportional to the number of sources (charges)!!! How intuitive is that?! From that idea everything follows so naturally! Ah brain explosion !









Bonus: What if it was a real surface? The constants will simply change! Some media are more resistant for this electric fluid to flow than others!