Calculating Electric Lines of Force for Arbitrary Charge Configurations

[Shan K]

can anyone tell me how to calculate the electric lines of force of an arbitrary electric charge configuration ?



i have heard that in static case there is no electric field inside a conductor . i now want to know that is there any dynamical case for the conductor instead of static ?

 

[andrien]

it can be very difficult to know the nature of field lines in general case.Even in two charge case,one use prolate ellipsoidal coordinate to draw equipotentials and determining the nature of force lines.In static case,there is no field inside a conductor but in presence of say a time varying magnetic field there can be an electric field inside it.

 

[Shan K]

it can be very difficult to know the nature of field lines in general case.Even in two charge case,one use prolate ellipsoidal coordinate to draw equipotentials and determining the nature of force lines.

but in a book i have seen it but i can't undetrstand . they taken that electric field is parallal to a small elementary distance dl . but i can't understant what dl is . the distance from whom and to whom ?

 

[tris_d]

but in a book i have seen it but i can't undetrstand . they taken that electric field is parallal to a small elementary distance dl . but i can't understant what dl is . the distance from whom and to whom ?

Distance is when you are talking about FORCE lines and two objects. For FIELD lines you only need one object, so I guess you are interested in field lines, but what kind of object: wire, surface or distribution of point charges?

 

[mishima]

From Chabay/Sherwood Electric and Magnetic Interactions:

Step 1: Cut up the charge distribution into pieces and draw E vector for one piece.
-Very small pieces can be approximated by point particles
-Pick out a representative piece, and at the location of interest (where do you want to find the field?) draw a vector E showing the contribution to the electric field of this representative piece. Drawing this vector helps you figure out the direction of the net field at the location of interest. (you are simply using Coloumb's law with one charge being the small piece, and the other you imagine as a positive point charge.)

Step 2: Write an expression for the electric field due to one piece
-invent an integration variable to refer to the various pieces. The integration variable will not appear in the final result, but you will need it to refer algebraically to one of your pieces.
-write algebraic expressions in terms of your integration variable for the vector components of E
-if your representative piece is infinitesimal in size, your integration variable must include infinitesimal increments of the integration variable. For example, if your integration variable is y your expressions must be proportional to delta y.

Step 3: Add up the contributions of all pieces
-Write an expression for the net field as the sum of the contributions of all the pieces. (this is allowable due to the superposition principle) If the individual contributions are infinitesimal, write the sum as a definite integral whose limits are given by the range of the integration variable. If the integral can be done symbolically, do it. If not, choose a finite number of pieces and do the sum with a calculator or computer. (excel is good enough for this)

Step 4: Check the result
-Check that the direction of the net field is qualitatively correct.
-Check units, which should be Newtons / coulomb
-look at special cases for which you already know the answer. For example, if you have some net charge, then at an extreme distance you should get something that looks like a single point charge.

I'd add that when you are doing step 2, use proportional reasoning instead of thinking about charge density and such. This allows you to derive charge density and is much more intuitive in my opinion. For example, say your object is a uniformly charged rod. Then it must be the case that

delta x / total length = total charge / delta charge

Delta charge is what goes in your expression for step 2.


As for your second question about conductors, its true inside a conductor in static equlibrium there is a net field of zero. There can also be surface charges on the metal if there is an external charge and yet still be zero inside during static equilibrium. This is called polarization. For the fraction of a second before static equlibrium is reached, the net electric field inside the metal is non-zero.

The "dynamical" case you are looking for is an electric circuit. Inside a conductor in this case there is current, and a non-zero electric field.