The cross-section area and length of a cylindrical conductor are A and l, respectively. The specific conductivity varies as σ(x) = σ 0 
, where x is the distance along the axis of the cylinder from one of its ends [see Fig.
Fig. Compute the resistance of the system along the cylindrical axis.
Text Solution
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Sol. Consider the cylinder as composed of thin discs of width dx connected in series [see Fig. ]. The resistance of a disc at a distance x away from the cylinder end is:
dR = 
= 
….. (1)
where A is the cross-section area of the disc and dx is width. Since the discs are connected in series, the total resistance is
R = 
= 
dx = 
….. (2)
Fig.
From Ohm's law, we deduce that the current flowing across the cylinder is given by
I = 
= 
….. (3)
The current density is, therefore:
J = 
= 
…. (4)
The electric field in the cylinder may be found by using Ohm's law:
E(x) = 
= 
= 
….. (5)
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