P.27 |
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P.28 Surfaces of Constant Potential |
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Statement
Given the potential function V as a product of two functions in two different variables, determine the shape and location of a surface on which V is constant (equipotential).
System Parameters
Solution
The solutions of Laplace's equation when it varies over two dimensions are often products of two equations in the two distinct variables, as shown in Chapter 8. Such a solution is given above for the function V(x,z). Since the potential is not a function of y, the equipotential surfaces extend to ±¥ in the y direction. Because sin(az) = 0 for both z = 0 and z = np/a, where n = 1, 2, 3, ..., the planes z = 0 and z = p/a are at a zero potential. Because sinh(ax) = 0 at x = 0, the plane x = 0 is also at zero potential. The V = 0 potential is shown in red on the graph below. For another equipotential surface, the following holds true:
When values of z between zero and p/a are substituted, the corresponding x coordinates are readily obtained.
Number of points to plot:
Create a reference range for each of the points to plot:
Create a value of z for each point:
The conditions on z are to prevent the function solver below from solving the equation for values of z which will cause x to be infinite.
Use the root function to find the values of x for which the above equation does equal zero. First enter a guess value for x.
Guess:
The red lines at x = 0, z = 0 and z = p/a are lines of zero potential. The blue, dotted line is the equipotential surface V(x,z) = V0, and the cyan, solid line is the surface V(x,z) = poten. Try changing the variable poten defined to the left of the graph. This will show the different equipotentials of voltage poten. Because V is periodic in z, and because V(-x,-z) = V(x,z), the whole xz plane can be filled with replicas of the strip shown in the graph.
You can easily calculate the electric field given this potential function. What would you expect the electric field to look like? Does the numerical solution agree with your physical intuition of how the field lines should appear? Remember that, in Chapter 5, it was asserted that the electric field lines are always perpendicular to the equipotential lines. Does that seem to be true?
