Question

Number of particles passing in unit time through unit area perpendicular to the z-axis is given by: 
$N = D \dfrac{(N_2 - N_1)}{(Z_2 - Z_1)}$, 
where $N_2$ and $N_1$ are the number of particles in unit volume at $Z_2$ and $Z_1$. The dimensional formula of $D$ is?

• (A) M⁰L^-2T²
• (B) M⁰L^-1T^-1
• (C) M⁰L²T^-1
• (D) M⁰L^-2T^-2

Answer 1

The Correct Option is C

Explanation:
Dimensional formula for number of particles per unit area per unit time is $\left[M^0{L}^{-2}{T}^{-1}\right]$${\mathrm{N}}_1$ and ${\mathrm{N}}_2$ are the number of the particles in unit volume. Dimensional formula for ${\mathrm{N}}_1$ is $\left[M^0{L}^{-3}{T}^0\right]$Dimensional formula for $z_1$ and $z_2$ is $\left[M^0{L}^1{T}^0\right]$ Therefore, dimensional formula for $D$$[D]=\frac{[N]\left[Z_1\right]}{\left[N_1\right]}$$[D]=\frac{\left[M^0{L}^{-2}{T}^{-1}\right][L]}{\left[M^0{L}^{-3}{T}^0\right]}=\left[M^0{L}^2{T}^{-1}\right]$

Answer 2

Principle of Homegeinity

According to the Principle of Homogeneity, the dimensions of each term in a dimensional equation on both sides should be the same.

To check the correctness of a given equation using dimensional analysis, we should apply the homogeneity principle to the equation.

For example, the given physical equation is

Kinetic energy, E = 1/2 mv²

Where m is the mass and v is the velocity.

The above equation will be dimensionally correct if the dimensions of the right side of the equation are the same as that of the left side of the equation.

Limitations of Dimensional Analysis

The limitations of dimensional analysis are

• Dimensionless constant involved in the physical relation cannot be determined.
• The dimensional method fails to derive the relation like

S = ut + 1/2 at² and v² - u² = 2aS

• It fails to give information about trigonometric functions, logarithmic, exponential, and complex quantities.
• A dimensionally correct relation may not be true physical relation.