Why in a small unit of area for a surface S in an electric field

I am wondering why in a small unit of area for a surface S in an electric field. The flux of the small area is the dot product of E and s where s is the normal vector and makes an angle theta to the field. Why do we use the dot product of E and s, why does it give us the flux of the small area?
 
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Visualize the electric field as "wind". The dot product tells you how much
"air" is passing through the small surface. Don't take the analogy any further
than that- it's just a good way to visualize flux and dot products.
 
You need to use the dot product because you need the amount "going in" to a surface. Imagine if you turn a can of coke on its side so that the circular ends are like vertical circles. Flux is the amount of whatever flowing through a given area. So like in the previous post, if the wind was perfectly vertical, nothing would enter the can from its ends. If we take the dot product of the winds direction with that of the outward normal from the ends caps we get 0. However, if the wind is perfectly horizontal we get some non zero value. (we actually need to take the negative of the dot product if we use the outward normal). That is if the outward normal of one end cap points to the left and the winds is to the right we would obtain a negative value for flux. This is incorrect as inward flux has a positive sign convention and outward flux has a negative sign convention. By taking the negative of left dot right we obtain some positive value which in this case is the magnitude of the air flow rate through the circular area of the left side of the can. In fluid dynamics flux is very important as the net flux determines whether or not a control volume maintains a steady state or not.
 

Gokul43201

Staff Emeritus
Science Advisor
Gold Member
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misogynisticfeminist said:
I am wondering why in a small unit of area for a surface S in an electric field. The flux of the small area is the dot product of E and s where s is the normal vector and makes an angle theta to the field. Why do we use the dot product of E and s, why does it give us the flux of the small area?
Simple answer : "Because that's how the flux is defined" !
 

Claude Bile

Science Advisor
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To emphasise Gokul's point;

Flux is maximum when E is parallel to the surface normal, and zero when these two vectors are perpendicular. It is therefore intuitive to use a dot product to define the flux.

Claude.
 

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