Question

The time dependence of a physical quantity p is given by $p = {p_0}\exp \left( { - \alpha {t^2}} \right)$, where $\alpha$ is constant t is the time. The constant $\alpha$
A. is dimensionless
B. has dimension ${T^{ - 2}}$
C. has dimension ${T^2}$
D. has dimension of p

Answer 1

The exponential term is always dimensionless in nature. Therefore, in the mentioned expression the total sum of the powers of dimensions of the physical quantity must be zero for the exponential term to be dimensionless. The physical quantities with a like or similar dimension can only be added or subtracted. The homogeneity is necessary for the addition or subtraction of the physical quantities. The dimension of the time is $\left[ {\text{T}} \right]$, mass is $\left[ M \right]$ and length is $\left[ L \right]$. All the dimensions of other quantities are derived from these basic dimensions.

Complete step-by-step solution :
Given- The time dependence of physical quantity is $p = {p_0}\exp \left( { - \alpha {t^2}} \right)$.
The equation to calculate the dimensions of constant is,
$\left[ \alpha \right]\left[ {{t^2}} \right] = \left[ {{{\text{T}}^0}} \right]$
The exponential term in the expression is the constant. The power of the constant term must be such that the algebraic sum of all the powers of time and the constant become zero.
Substitute the dimensions in the above expression.
$\left[ {{{\text{T}}^x}} \right]\left[ {{{\text{T}}^2}} \right] = \left[ {{{\text{T}}^0}} \right]$
$\left[ {{{\text{T}}^{x + 2}}} \right] = \left[ {{{\text{T}}^0}} \right]$
$x + 2 = 0$
$x = - 2$
The power of the constant term is -2, so the dimension of the constant term $\alpha$ has an inverse dimension of the square of the time.
Thus, the constant $\alpha$ has the dimensions ${T^{ - 2}}$ and the option (B) is correct.

Note:- Remember that the dimensions in the product of the physical quantities must be balanced. The dimension of a physical quantity in the product of two physical quantities must be inverse to the dimension of another physical quantity. The principle of dimensional homogeneity must be used to calculate the dimensions of the unknowns in an expression.