Semi-infinite Plate
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 59
11-8-2016
1435
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 xi 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
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