Semi-infinite Plate

Consider a thin semi-infinite plate of negligible thickness made of an isotropic conductive material. A voltage V₀ = 1 V is applied across points A and B of the plate (see Figure 1.1). At a distance d = 1 cm from the end a voltage of 0.1 V is measured between points C and D. Find the voltage  difference between two analogous points an arbitrary distance x from the end of the plate.

Figure 1.1

SOLUTION

Consider the voltage difference between arbitrary points (E, F) a distance x_i from the end (see Figure 1.2). We can write

(1)

Since the plate is semi-infinite, (1) is true for any other points (G, H) of the plate. So

(2)

Figure 1.2

Hence, for an arbitrary point a distance from the end, we have

(3)

or

(4)

(5)

From the condition V(d) = V₀/10, we have

(6)

Therefore

For the values given in the problem, the voltage measured a distance x cm from the end gives the function