Problem: A new conducting Ohm\'s law obeying material has been developed with a

Question

Problem: A new conducting Ohm's law obeying material has been developed with a position-dependent resistivity, , such that (x) = B + Cx , B and C are positive constants, and x is the position of an infinitesimal piece of the material measured along an x-axis (x 2 0). The material is used to construct a right cylindrical solid resistor having constant cross-sectional area A and length L such that x = 0 at one circular face and x = L at the other face. An ideal battery having terminal voltage V is connected across the resistor such that the positive terminal is connected at x = L and the negative terminal is connected at x 0 a) What are the SI units of the constants B and C? Determine expressions (in terms of quantities described above) for b) the total resistance of the resistor; c) the current flowing through the resistor; d) the electric field in the resistor at x = 2; and e) the potential in the resistor at x = L/2 (relative to the negative terminal of the battery)

Explanation / Answer

a) si unit of B= ohm.m

Si unit of C = ohm

b) R= rho.dx/A from x= 0 to L

(B+ Cx).dx /A = (BL + CL^2/2) ( ans)

c) I = V/(BL + CL^2/2) ( ans)

d) E = I*Rho/A = [{V/(BL + CL^2/2) * (B+Cl/2)}/A] ( ans)

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