Physics

Consider the following two statements A and B. Identify the correct answer.
A) The quantity $$\dfrac { { e }^{ 2 } }{ { \epsilon  }_{ 0 }ch } $$ is dimensionless.
B) $$\dfrac { 1 }{ \sqrt { { \mu  }_{ 0 }{ \epsilon  }_{ 0 } }  } $$ 
has the dimensions of velocity and is numerically equal of velocity of light.


ANSWER

Both A and B are true


SOLUTION
Dimensions of permittivity $${ \epsilon  }_{ 0 }$$: $${ A }^{ 2 }{ T }^{ 4 }{ M }^{ -1 }{ L }^{ 3 }$$
Dimensions of $${ e }^{ 2 }$$ : $${ A }^{ 2 }{ T }^{ 2 }$$
Dimensions of $$c$$ : $$L{ T }^{ -1 }$$
Dimensions of $$h$$ : $$M{ L }^{ 2 }{ T }^{ -1 }$$
Therefore, dimensions of the given expression:
$$\dfrac { { A }^{ 2 }{ T }^{ 2 } }{ ({ A }^{ 2 }{ T }^{ 4 }{ M }^{ -1 }{ L }^{ 3 })\times (L{ T }^{ -1 })\times (M{ L }^{ 2 }{ T }^{ -1 }) } =1$$ which is dimensionless.

Hence, Statement A is true.
In statement B, the expression given is for the speed of light.Hence, it has dimensions of velocity.
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