Question

If L , C and R denote the inductance, capacitance and resistance respectively. The dimensional formula for C²LR is :
$\bold{a) \: \:[M{L}^{ - 2} {T}^{ - 2}{ I }^{0} ] }$
$\bold{b) \: \:[{M }^{0} {L }^{0} {T }^{3}{ I }^{0} ] }$
$\bold{c) \: \: [{M}^{ - 1}{ L}^{ - 2} {T }^{6} {I}^{2} ]}$
$\bold{ b) \: \:[{M}^{0} {L }^{0} {T }^{2} {I }^{0} ] }$

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Answer 1

$\bold{b) \: \:[{M }^{0} {L }^{0} {T }^{3}{ I }^{0} ] }$

We know,

$\sf{[L]=[M{L}^{2}{T}^{-2}{I}^{-2}]}$

$\sf{[C]=[{M}^{-1}{L}^{-2}{T}^{4}{I}^{2}]}$

$\sf{[R]=[M{L}^{2}{T}^{-3}{I}^{-2}]}$

So, the dimensional formula for

$\sf{[{C}^{2}LR]}$

$\sf{=[{({M}^{-1}{L}^{-2}{T}^{4}{I}^{2})}^{2}(M{L}^{2}{T}^{-2}{I}^{-2})(M{L}^{2}{T}^{-3}{I}^{-2})] }$

$\sf{=[({M}^{-2}{L}^{-4}{T}^{8}{I}^{4})(M{L}^{2}{T}^{-2}{I}^{-2})(M{L}^{2}{T}^{-3}{I}^{-2})]}$

$\sf{=[({M}^{-2+1+1})({L}^{-4+2+2})({T}^{8-2-3})({I}^{4-2-2})]}$

$\sf{=[{M}^{0}{L}^{0}{T}^{3}{I}^{0}]}$