Calculating Area Vectors in Gauss' Law

In summary, to find the direction of the area vector of a surface, there are a few ways to calculate the little area element. For a closed surface, the direction is from inside to outside for the positive direction and vice versa for the negative direction. For an open surface, the direction is determined by following the right-hand rule while traversing the boundary. Additionally, if the surface is given as an equation, evaluating the gradient of the equation at a chosen point will give a vector normal to the surface, which can then be normalized to find the unit vector ##\hat{n}## and the little area element ##d\vec{S} = \hat{n}dA##. Another method involves using curvilinear coordinates and the
  • #1
Prashasti
63
2
How do I find the direction of area vector of a surface?
 
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  • #2
For a closed surface it goes from INSIDE to OUTSIDE for the positive direction, and the reverse for the negative direction.

For an open surface you will follow an arbitrary convention which is based upon a traversal of the boundary. Then you follow the right-hand rule: if your fingers wrap around the boundary, your thumb points in the positive direction.
 
  • #3
I agree. Also, to actually calculate the little area element ##d\vec{S}## there are a few ways. Firstly, if you are given the surface as an equation of the form ##f(x,y,z)=0## then
[tex]\nabla (f)[/tex]
evaluated on the surface, will give you a vector normal to the surface, at the point you choose. Of course, you will still need to normalise this vector, and choose which way is 'outwards'. So then you will have a unit vector ##\hat{n}## and so your little area element ##d\vec{S} = \hat{n}dA##

Another nice way, is if you are given general curvilinear coordinates on the surface (call them ##u## and ##v##), and if you know ##\vec{r}(u,v)## i.e. the position in 3d space, as a function of ##u## and ##v##, for all positions which lie on the surface. Then the little area element is:
[tex]d\vec{S} = \left( \frac{\partial \vec{r}}{\partial u} \wedge \frac{\partial \vec{r}}{\partial v} \right) \ du \ dv [/tex]
 
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1) What is the area vector in Gauss' Law?

The area vector in Gauss' Law is a vector that points perpendicular to the surface element of a closed surface. It is used to calculate the flux through a closed surface in order to determine the electric field.

2) How is the area vector related to electric flux?

The area vector is directly related to electric flux through a closed surface. The electric flux is equal to the dot product of the electric field and the area vector. This means that the magnitude of the electric flux is equal to the magnitude of the electric field multiplied by the magnitude of the area vector and the cosine of the angle between them.

3) How is the direction of the area vector determined?

The direction of the area vector is determined by the right-hand rule. If you curl the fingers of your right hand in the direction of the closed surface's normal vector, then your thumb will point in the direction of the area vector. This direction is important in determining the sign of the electric flux, and therefore the direction of the electric field.

4) Can the area vector be negative?

No, the area vector is a physical quantity and cannot be negative. However, depending on the orientation of the closed surface and the direction of the electric field, the dot product of the two vectors may result in a negative value for the electric flux.

5) How is the area vector used in Gauss' Law calculations?

The area vector is used in Gauss' Law calculations to determine the electric flux through a closed surface. By calculating the dot product of the electric field and the area vector, the electric flux can be determined, which can then be used to solve for the electric field at a specific point.

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