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A rare case in which the electrostatic field E for a
Chapter 7, Problem 42P(choose chapter or problem)
Problem 42P
A rare case in which the electrostatic field E for a circuit can actually be calculated is the following:28 Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the battery) is maintained at and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm’s law, then,
a) Use separation of variables in cylindrical coordinates to determine V (s, φ) inside and outside the cylinder.
b) Find the surface charge density on the cylinder.
Questions & Answers
QUESTION:
Problem 42P
A rare case in which the electrostatic field E for a circuit can actually be calculated is the following:28 Imagine an infinitely long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the battery) is maintained at and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm’s law, then,
a) Use separation of variables in cylindrical coordinates to determine V (s, φ) inside and outside the cylinder.
b) Find the surface charge density on the cylinder.
ANSWER:
a.)
Step 1 of 4
We have to determine the potential inside and outside the cylinder of uniform resistivity
and radius in cylindrical coordinates.
The general solution for the potential with cylindrical symmetry is
So, the potential inside and outside the cylinder is
The cosine terms is not considered because is an odd function of .
At ,