Help compute integral (electric potential)

In summary, the conversation discusses how to calculate a given integral and how to integrate it over a plane in R^3 using Stokes Theorem. It is suggested to use a perspective or orthographic projection to transform the plane onto a 2D space and then use the surface integral relationship to calculate the area. The process involves dividing the surface into small points and finding the normal vector at each point, then using the dot product with the standard base normal to calculate the surface area.
  • #1
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Homework Statement



How to calculate the following integral:

Integral( 1/ |x-y| dS) where x and y are vectors in R^2, || represents norm.
So say x= (x1,x2), y=(y1,y2). Then the integral is [ dS/ sqrt ((y1-x1)^2 + (y2-x2)^2)].

The problem is: I need to integrate this over any plane in R^3.

The Attempt at a Solution



So I assume the plane is of the form ax+by+cz+d =0. Then I really have no clue how to proceed.
 
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  • #2
Hi Carl140! :smile:

(have an integral: ∫ and a square-root: √ and try using the X2 tag just above the Reply box :wink:)
Carl140 said:
Integral( 1/ |x-y| dS) where x and y are vectors in R^2, || represents norm.
So say x= (x1,x2), y=(y1,y2). Then the integral is [ dS/ sqrt ((y1-x1)^2 + (y2-x2)^2)].

The problem is: I need to integrate this over any plane in R^3.

Probably easiest if you let P be the closest point on the plane to the origin, and use coordinates based on P. :wink:
 
  • #3
tiny-tim said:
Hi Carl140! :smile:

(have an integral: ∫ and a square-root: √ and try using the X2 tag just above the Reply box :wink:)


Probably easiest if you let P be the closest point on the plane to the origin, and use coordinates based on P. :wink:

Thanks but how should I choose the limits of integration and how to calculate the area element? I'm very confused with this part.
 
  • #4
By itself, the integral doesn't make any sense. What are the variables of integration? Are you integrating over the vector x or y or are both vectors a function of the position on the plane? This would help determine the area element. As for the limits of integration, that's up to what you are trying to calculate.

You say the vectors are in R^2 space but you want to integrate over R^3. How do these vectors lie in the R^3 space?
 
  • #5
What you want to do is use Stokes Theorem by first projecting the plane to a two dimensional plane (that is the XY plane for example) and then you use what you call the surface integral relationship which is given by

What you need to picture is the projection of a three dimensional plane onto a 2D space.

I'll just say a few things about projections and how they work. There are two major types of projections:

The first one is the perspective projection and the second major class is the orthographic projection.

The perspective projection essentially transforms the x,y,z coordinate system to the xy plane in the following manner:

x' = x/z
y' = y/z

Note here that we can't use z = 0 because the result will be infinite but essentially it scales the coordinates so that we get a feeling of "depth".

If we have say the points of a cube and we transform these eight points into the XY plane, then we get the following result:

Initial points:

Front face of cube
-1,+1,+1
-1,-1,+1
+1,-1,+1
+1,+1,+1

Back face of cube
-1,+1,+2
-1,-1,+2
+1,-1,+2
+1,+1,+2

Now we take the perspective transformation on all points to the x,y plane by using the transformations listed above.

Transforming Front face of cube

(-1,+1,+1) - > (-1,1)
(-1,-1,+1) - > (-1,-1)
(+1,-1,+1) - > (1,-1)
(+1,+1,+1) - > (1,1)

Transform Back face of cube
(-1,+1,+2) -> (-1/2,1/2)
(-1,-1,+2) -> (-1/2,-1/2)
(+1,-1,+2) -> (1/2,-1/2)
(+1,+1,+2) -> (1/2,1/2)

If you draw these points on a paper with a cartesian coordinate system you will see that the points corresponding to the larger z coordinate are smaller which means that as
distance increases you get the illusion of further distance. Take note however this is
a simple projective transformation. Other transformations can exist that are nonlinear and not of this type but I've just decicided to illustrate a simple example.

As for orthographic transformations these potentially should preserve length in a coordinate system. So what you want to do is use orthographic projections in place of perspective when talking about projecting the surface to the XY plane.

Now we have to explain how the surface integral works.

This is a rough sketch of how to explain the surface integral.

If we divide the surface up into extremely small "bits" then we get a normal vector at each point. Now what we do is if we picture the surface to be a collection of "points" with a normal vector at each point, then what we are picturing is basically a relative
measure (ie another form of saying the angular measure) between the standard normal (ie 0,0,1) and the surface normal.

The contribution is given by the inner product (or just dot product) of the surface normals with the standard base normal ie (0,0,1). If we sum these components and multiply by the projected area, then we have what the call the surface area.

Hope that helps.
 

1. What is an electric potential?

An electric potential is a scalar quantity that represents the amount of work needed to move a unit charge from a reference point to a specific point in an electric field. It is often denoted by the symbol V and is measured in volts (V).

2. What is the significance of computing an electric potential?

Computing an electric potential is important in understanding the behavior of electric fields and how they affect charged particles. It can also help in determining the direction and strength of the electric field in a given region.

3. How is the electric potential related to the electric field?

The electric potential is related to the electric field by the equation V = -∫E⋅dl, where V is the electric potential, E is the electric field, and dl is the infinitesimal displacement along the path of integration. In other words, the electric potential is the negative of the line integral of the electric field.

4. What are the different methods for computing an electric potential?

The most common methods for computing an electric potential include using the Coulomb's law, the principle of superposition, and using Gauss's law. These methods involve mathematical calculations and can be done analytically or numerically, depending on the complexity of the electric field.

5. Can the electric potential be negative?

Yes, the electric potential can be negative. This indicates that the work done to move a unit charge from the reference point to the specific point is negative, meaning that energy is being released rather than absorbed. This can occur in regions where the electric field is directed towards the reference point.

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