2. Plotting Field Lines and Equipotentials

• Review: Ingredients for the field-line plotter:

  • a function to compute the potential phi(x,y)
  • a function to compute the electric field E(x,y)
  • a function to take a step of specified length in the local direction of, or opposite to, the E-field
  • a function to choose (place limits on) the step
  • a function to set up the charge distribution
  • a function to draw the field line through a specified point, stopping the calculation on a charge or when the line is too far from the origin
  • a function to draw all the field lines originating near any charge
  • a main program to set parameters and oversee the calculation

• Exercise 2.1:  Compute the field-line pattern due to the dipole in problem 1.2, choosing starting points and ending criteria as discussed in class.  [Solution]

• Exercise 2.2:  Now let's try a configuration that is a little harder to guess! Modify your program to compute the field-line pattern due to the following distribution of charges:

  	nq = 4
  	q1 =  2.e-6,  x1 =  0,  y1 =  0
  	q2 = -1.e-6,  x2 = -1,  y2 =  1
  	q3 = -2.e-6,  x2 =  1,  y2 =  1
  	q4 = -1.e-6,  x2 =  0,  y2 = -0.73

   [Solution]

• Equipotentials can be handled somewhat similarly (now taking a step in the direction perpendicular to the local field), except that:

  • The choice of initial conditions is obviously different.

  • The major change in actually taking a step is that, instead of

    	dx = direction*ds*Ex/E
    	dy = direction*ds*Ey/E

    we now have

    	dx = -direction*ds*Ey/E
    	dy = direction*ds*Ex/E

  • At the end of every step, we now have an explicit measure (dphi = phi - phi_initial) of how far we are from the correct equipotential, so we can apply a correction at the end of each step to force us back onto the correct track. Operationally, at the end of the step we compute dphi, the difference between the current potential and the correct value, then take a corrector step of length ds along the direction of the local electric field (Ehat) so as to change the potential by -dphi: i.e. (ds Ehat).(E Ehat) = dphi, so ds = dphi/E. (Thus, if the calculated potential is too high, we take a step along the field line to reduce it; if it is too low we take a step in the opposite direction.)

  • The equipotentials form closed loops, so special care must be taken to make the loops actually close:
    • use a symmetric integrator, which may be able to reduce the error to near machine precision
    • stop when "near" the starting point; specifically, monitor the distance from the current point on the equipotential to the starting point, and stop when that distance has a minimum less than some tolerance -- 0.01, say (the detailed value may need some fine tuning)

  Demo:  Plotting equipotentials one by one.

• Exercise 2.3:  For the dipole in Exercises 1.2 and 2.1, compute the equipotentials through (x, y) = (0, Y), for Y = -0.9, -0.8, -0.7, ..., 0.9.  [Solution]

• Exercise 2.4:  For the charge configuration in Exercise 2.2, compute the equipotentials through (x, y) = (0.5, Y) and (-0.5, Y), for Y = -4.0, -3.0, ..., 4.0.  [Solution]  [Solution (random)]  [Interactive solution]