Re: Electric Fields

Note: I've included two figures in this answer which show the sort of graphics your program should produce. I don't want to take the wind out of your sails by suggesting I've already written your program: I used a commercial software package called 'MATLAB' to generate these figures, and I haven't the faintest idea how to write a program which produces them -- that's your job.

This formula gives the force on a charge Q1 due to the presence of a second charge, Q2. The field is defined as the force per unit charge on a tiny "test charge" dQ in the presence of a charge Q: using the same formula above, but changing notation,

F = k dQ Q /d2

     F     k Q
E = --- = ----- 
     dq     d2

Three more things you need to know:

1. The electric field E is a vector. It has a magnitude and a direction. The equation above is for the magnitude; the direction is always away from the charge Q (for positive Q). You have to draw the electric field as an arrow pointing toward or away from the charge. The x and y components of this vector can be written

        F     k Q x 
   Ex = --- = ----- 
        dq     d3
   
        F     k Q y 
   Ey = --- = ----- 
        dq     d3
   

   where x is the horizontal distance from the charge, y is the vertical distance, and d = sqrt(x^2+y^2) is the total distance. E_x is a vector which points in the x direction (toward the right), and E_y is a vector pointing in the y direction (upward). The total field is the sum of these two vectors -- for example, if E_x is positive (rightward) and E_y is negative (downward), the total E vector points down and to the right.
2. The electric field of several point charges equals the sum of the electric fields of each. You can compute E_x and E_y for each charge separately, and add them up at the end. This is done for two charges in the figure below: notice that the arrows don't point exactly at one charge or another!
   
3. The concept of an electric potential is useful. The potential is a scalar (not a vector) whose gradient gives the electric field. If you don't know calculus, that last sentence didn't make much sense: I'll just say that the electric potential makes it easy to calculate the electric field. The electric potential of a point charge is given by

          k Q
   V = - -----
           d
   

   (note only one d in the denominator!) If you know the potential at every point on a grid V(i,j), you can calculate the electric field everywhere this way:

      Ex = (V(i+1,j) - V(i-1,j))/(2*deltax)
      Ey = (V(i,j+1) - V(i,j-1))/(2*deltax)
   

   that is, the vertical component of E is given by taking the difference of V at points above and below the point of interest and dividing by twice the separation between gridpoints, and the horizontal component is given by taking the difference between points to left and right. Watch out for divide-by-zero!

   This is the technique I used to generate the first figure. Another useful graphic is to display contours of the potential itself: the electric field is strongly positive near positive charges and negative near negative charges: the electric field is perpendicular to the contours of the potential, and the strength of the field is proportional to the contour density. Below, I've shown a contour-plot of the electric potential superimposed on a color-graphic of the potential. Red is positive potential, blue is negative.