Question
The time dependence of physical quantity $$p$$ is given by $$p = {p_0}\exp \left( { - \alpha {t^2}} \right),$$ where $$\alpha $$ is a constant and $$t$$ is the time. The constant $$\alpha $$
A.
is dimensionless
B.
has dimensions $$\left[ {{T^{ - 2}}} \right]$$
C.
has dimensions $$\left[ {{T^2}} \right]$$
D.
has dimensions of $$p$$
Answer :
has dimensions $$\left[ {{T^{ - 2}}} \right]$$
Solution :
$$p = {p_0}\exp \left( { - \alpha {t^2}} \right)$$
As powers of exponential quantity is dimensionless, so $${ \alpha {t^2}}$$ is dimensionless.
$$\eqalign{
& {\text{or}}\,\alpha {t^2} = {\text{dimensionless}} = \left[ {{M^0}{L^0}{T^0}} \right] \cr
& \therefore \alpha = \frac{1}{{{t^2}}} = \frac{1}{{\left[ {{T^2}} \right]}} = \left[ {{T^{ - 2}}} \right] \cr} $$