A truncated right circular cone, made out of insulator, has the height h, base radius a 1 and top radius a 2 (a 2 < a 1 ). The base sits on a conducting plate and the top supports a cylindrical conducting rod of radius a 3 , such that a 3 < a 2 (Fig.). If the surface of the cone has a surface resistivity ρ S , then show that the surface resistance between the plate and the rod is
Text Solution
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Sol. r = radius of a circular disc and R = the corresponding cone generator radius.
∴ 
= sinα ..... (1)
Where α is semi-vertical angle of the cone.
∴ α = tan –1 
Hence, 
= dR ..... (2)
We consider a circular section of the cone of radius r.
∴ The elemental strip is of inclined height dR.
∴ Surface resistance of the element,
dR S1 = 
= 
..... (3)
Hence the total surface resistance of the inclined part of the cone,
R S1 = 
= 
= ln 
.... (4)
We have also to consider the surface resistance of the annular ring of the top end from r = a 3 to r = a 2 .
∴ R S2 = 
= 
= 
ln 
... (5)
∴ R S = R S1 + R S2 = 
where sin
= 
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