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A long roundconductorto cross-sectionalarea S is made of material whoseresistivitydepends onlyon a distance r fromthe axis of the conductoras rho = alpha//r^(4), where alpha is a constant. Find : (a) the resistance per unit lengthof sucha conductor, (b) the electric field strengthin the conductordue to which a current I flow though it. |
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Answer» Solution :(a) Consider a cylinder of UNIT length and divide it into shellsof radius `r` and thickness `dr` Different sections are in parallel. For a typical section, `d((1)/(R_(1))) = (2pi r dr)/((alpha//r^(2))) = (2pi r^(3) dr)/(alpha)` Intergating, `(1)/(R_(1)) = (piR^(4))/(2 alpha) = (S^(2))/(2pi alpha)` or,`R_(1) = (2pi alpha)/(S^(2))`, where `S = pi R^(2)` (b) Suposse the electric field inside is `E_(x)= E_(z)` (Z axis is alongthe axis of the conductor). This electric field cannot dependon `r` in steadyconditionswhen other componetsof `E` are absent, otherwiseone violates the circulation theorem `oint vec(E) . vec(dr) = 0` The CURRENT through a sectionbetweenradii `(r + dr, r)` is `2pi r dr (1)/(prop//r^(2)) E = 2pi r^(3) dr (E)/(prop)` Thus `I = int_(0)^(R) 2pi r^(3) dr (E)/(prop) = (pi R^(4) E)/(2 prop)` HENCE `E = (2 OO pi L)/(S^(2))` when `S = R^(2)` |
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