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09 April 2004

Charges on a Surface

George E. Hrabovsky, President MAST

News from MAST

MAST will be offering a severe weather spotter's base operations course beginning next month. This course will be free of charge and available over the Web.

Charged Curves on a Surface

We have seen how charged curves produce electric fields in one dimension.  Now we examine them in two dimensions.  Again, we have

E (r) = k∫_Cℓ (r)/r^2Overscript[r,^] l

for an electric field in one dimension.  We can take the line integral with respect to the surface element A,

E (r) = k∫_Cℓ (r)/r^2Overscript[r,^] A

Let us again assume that we have a unit of charge per unit of length. The electric field is then,

E (r) = k∫_Cℓ (r)/r^2Overscript[r,^] A

          = k∫_C1/r^2Overscript[r,^] A

From the Pythagorean theorem we know that

r^2 = x^2 + y^2 .

If we assume that our linear charge is along the x axis only, then we have

r^2 = y^2 .

So,

E (r) = (k x - k y)/y^2 .

We can plot this,

[Graphics:HTMLFiles/index_10.gif]

This is the general procedure for finding an electric field in a plane.

Theory Challenge from Last Time

If we have a vector described on a surface,

  a = (sin t, cos t),

we integrate it over the surface (where Overscript[N,^] is the unit normal vector),

∫∫a · A = ∫∫a · Overscript[N,^] A

To find this we need to determine the unit normal vector, which requires the unit tangent vector,

Overscript[T,^] = (a(t)/t)/(|| a(t)/t ||)

     = ( (sin t, cos t)/t)/(||  (sin t, cos t)/t ||)

     = (cos t, -sin t)/(|| (cos^2t + sin^2t)^(1/2) ||)

     = (cos t, -sin t) .

From this we can find the unit normal vector,

Overscript[N,^] = (Overscript[T,^]/t)/(|| Overscript[T,^]/t ||)

      = ( (cos t, -sin t)/t)/(||  (cos t, -sin t)/t ||)

      = (-sin t, -cos t)/(|| (cos^2t + sin^2t)^(1/2) ||)

      = (-sin t, -cos t) .

So, the surface integral becomes,

∫∫a · A = ∫∫a · Overscript[N,^] A

               &nbs ...         = ∫∫ (-sin t sin t - cos t cos t) A

                         = ∫∫ -1A

                         = ∫ -xy

                         = -x y .

 

Theory Challenge

How would you integrate a function over a space of three dimensions?

Sources That I Like

Books:

Robert C. Wrede (1963), Introduction to Vector and Tensor Analysis, John Wiley and Sons (republished by Dover Publications in 1973). This is my all-time favorite book on vectors.

Edward M. Purcell (1985), Electricity and Magnetism, McGraw-Hill (Volume 2 of the famous and excellent Berkeley Physics Course). This is a very good introduction to field theory.

David J. Griffiths (1999), Introduction to Electrodynamics, Prentice-Hall. An extremely thorough treatment of field theory applied mostly to electrodynamics. Covers vector and tensor analysis along with special functions, complex analysis (conformal mappings), and relativity.

Online:

This first web page is pretty nice for getting a feel for vector fields.

http://www.math.duke.edu/education/ccp/materials/mvcalc/vfield/

A more advanced treatment can be found here,

http://vishnu.mth.uct.ac.za/omei/a-calculus/chap4/node2.html

Created by Mathematica  (April 8, 2004)

 
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