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| 1. |
A long round conductor of cross- sectional area S is made of material whose resistivity depends only on distance r from the axis of the conductor as rho = (alpha)/(r^2) , where alpha is a constant. Find (a) the resistance per unit length of such a conductor. (b) the electric field strength in the conductor due to which a current i flows through it . |
| Answer» <html><body><p></p>Solution :Consider a <a href="https://interviewquestions.tuteehub.com/tag/cylindrical-429099" style="font-weight:bold;" target="_blank" title="Click to know more about CYLINDRICAL">CYLINDRICAL</a> element of radii between r and (r+ <a href="https://interviewquestions.tuteehub.com/tag/dr-959219" style="font-weight:bold;" target="_blank" title="Click to know more about DR">DR</a>) . Its resistance <br/> <img src="https://doubtnut-static.s.llnwi.net/static/physics_images/AKS_ELT_AI_PHY_XII_V02_A_C04_SLV_006_S01.png" width="80%"/> <br/> `dR = (rhol)/(2pi dr) ` (or)` <a href="https://interviewquestions.tuteehub.com/tag/1-256655" style="font-weight:bold;" target="_blank" title="Click to know more about 1">1</a>/(dR) = (2pi r dr)/(rhol)""……(i)` <br/>`<a href="https://interviewquestions.tuteehub.com/tag/therefore-706901" style="font-weight:bold;" target="_blank" title="Click to know more about THEREFORE">THEREFORE</a> 1/R = int_0^a 1/(dR) = int_0^a (2pi)/(rhol)r dr` <br/>(where a is the radius of the conductor ) <br/>`int_0^a (2pir dr)/(((alpha)/(r^2))l) = (2pi)/(alphal) int_0^a r^3 dr = (2pi)/(alpha l) ((a^4)/(4)) = ((pia^2)^2)/(2pialphal) = (S^2)/(2pi alpha l)` <br/>`R = (2pi alpha l)/(S^2) "".....(ii)` <br/>The resistance per unit length of wire `R= (2pi alpha)/(S^2)` <br/>(b) Equation (ii) can be written as `R= ((2pialpha)/(S)) ((l)/(S))` <br/><a href="https://interviewquestions.tuteehub.com/tag/compare-11929" style="font-weight:bold;" target="_blank" title="Click to know more about COMPARE">COMPARE</a> with `R = (rhol)/(S) , ` we get `rho = (2pi alpha)/(S)` <br/>By Ohm.s law `E= jrho = i/S xx (2pi alpha)/(S) = (2pi alpha i)/(S^2)`</body></html> | |