JEE MAIN - Physics (2016 - 9th April Morning Slot - No. 20)
In the following ‘I’ refers to current and other symbols have their usual meaning.
Choose the option that corresponds to the dimensions of electrical conductivity :
ML$$-$$3 T$$-$$3 I2
M$$-$$1 L3 T3 I
M$$-$$1 L$$-$$3 T3 I2
M$$-$$1 L$$-$$3 T3 I
Explanation
We know. resistivity ($$\rho $$) = $${{RA} \over L}$$
and conductivity = $${1 \over \rho }$$ = $${1 \over {RA}}$$
As R = $${V \over {\rm I}}$$
$$ \therefore $$ conductivity = $${{L{\rm I}} \over {VA}}$$
Also V = $${\omega \over q}$$ = $${\omega \over {it}}$$ = $${{\left[ {M{L^2}{T^{ - 2}}} \right]} \over {\left[ {\rm I} \right]\left[ T \right]}}$$ = $$\left[ {M{L^2}{T^{ - 3}}{{\rm I}^{ - 1}}} \right]$$
$$ \therefore $$ Conductivity = $${{\left[ {\rm{L}} \right]\left[ {\rm I} \right]} \over {\left[ {M{L^2}{T^{ - 3}}{{\rm{I}}^{ - 1}}} \right]\left[ {{L^2}} \right]}}$$
= $$\left[ {{M^{ - 1}}{L^{ - 3}}{T^3}{{\rm I}^2}} \right]$$
and conductivity = $${1 \over \rho }$$ = $${1 \over {RA}}$$
As R = $${V \over {\rm I}}$$
$$ \therefore $$ conductivity = $${{L{\rm I}} \over {VA}}$$
Also V = $${\omega \over q}$$ = $${\omega \over {it}}$$ = $${{\left[ {M{L^2}{T^{ - 2}}} \right]} \over {\left[ {\rm I} \right]\left[ T \right]}}$$ = $$\left[ {M{L^2}{T^{ - 3}}{{\rm I}^{ - 1}}} \right]$$
$$ \therefore $$ Conductivity = $${{\left[ {\rm{L}} \right]\left[ {\rm I} \right]} \over {\left[ {M{L^2}{T^{ - 3}}{{\rm{I}}^{ - 1}}} \right]\left[ {{L^2}} \right]}}$$
= $$\left[ {{M^{ - 1}}{L^{ - 3}}{T^3}{{\rm I}^2}} \right]$$
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