Question 339: Current Electricity

Question

Two conductors have the same resistance at $${0^ \circ }C$$ but their temperature coefficients of resistance are $${\alpha _1}$$ and $${\alpha _2}.$$ The respective temperature coefficients of their series and parallel combinations are nearly

Accepted Answer

$${R_1} = {R_0}\left[ {1 + {\alpha _1}\Delta t} ight];$$
$${R_2} = {R_0}\left[ {1 + {\alpha _2}\Delta t} ight]$$
$$R = {R_1} + {R_2}$$
$$ = {R_0}\left[ {2 + \left( {{\alpha _1} + {\alpha _2}} ight)\Delta t} ight]$$
$$ = 2{R_0}\left[ {1 + \left( {{{{\alpha _1} + {\alpha _2}} \over 2}} ight)\Delta t} ight]$$
$${\alpha _{eq}} = {{{\alpha _1} + {\alpha _2}} \over 2}$$
In Parallel, $${1 \over R} = {1 \over {{R_1}}} + {1 \over {{R_2}}}$$
$$ = {1 \over {{R_0}\left[ {1 + {\alpha _1}\Delta t} ight]}} + {1 \over {{R_0}\left[ {1 + {\alpha _2}\Delta t} ight]}}$$
$$ \Rightarrow {1 \over {{{{R_0}} \over 2}\left( {1 + {\alpha _{eq}}\Delta t} ight)}} = {1 \over {{R_0}\left( {1 + {\alpha _1}\Delta t} ight)}} + {1 \over {{R_0}\left( {1 + {\alpha _2}\Delta t} ight)}}$$
$$2\left( {1 - {a_{eq}}\Delta t} ight) = \left( {1 - {\alpha _1}\Delta t} ight)\left( {1 - {\alpha _2}\Delta t} ight)$$
$$	herefore$$ $${\alpha _{eq}} = {{{\alpha _1} + {\alpha _2}} \over 2}$$

Answer

$${{{\alpha _1} + {\alpha _2}} \over 2},\,{\alpha _1} + {\alpha _2}$$

Answer

$${\alpha _1} + {\alpha _2},\,{{{\alpha _1} + {\alpha _2}} \over 2}$$

Answer

$${\alpha _1} + {\alpha _2},\,{{{\alpha _1}{\alpha _2}} \over {{\alpha _1} + {\alpha _2}}}$$

Answer

$${{{\alpha _1} + {\alpha _2}} \over 2},\,{{{\alpha _1} + {\alpha _2}} \over 2}$$

Current Electricity

Two conductors have the same resistance at ${0^ \circ }C but their temperature coefficients of resistance are {\alpha _1} and {\alpha _2}.$ The respective temperature coefficients of their series and parallel combinations are nearly

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