Question

The equation of the stationary wave is
$y=2A\sin \left(\frac{2\pi ct}{\lambda }\right)\cos \left(\frac{2\pi x}{\lambda }\right)$
Which of the following statements is wrong?

• A. The unit of $ct$ is same as that of $\lambda$
• B. The unit of $x$ is same as that of $\lambda$
• C. The unit of $\frac{2\pi c}{\lambda }$ is same as that of $\frac{2\pi x}{\lambda t}$
• D. The unit of $\frac{c}{\lambda }$ is same as that of $\frac{x}{\lambda }$

Answer 1

The correct option is D The unit of $\frac{c}{\lambda }$ is same as that of $\frac{x}{\lambda }$

Here, $\left(\frac{2\pi ct}{\lambda }\right)$ as well as $\left(\frac{2\pi x}{\lambda }\right)$ are dimensionless. So, unit of $ct$ is same as that of $\lambda$. Unit of $x$ is same as that of $\lambda$. Also, $\left[\frac{2\pi ct}{\lambda }\right]=\left[\frac{2\pi x}{\lambda }\right]=M^0{L}^0{T}^0$
Hence, $\left[\frac{2\pi c}{\lambda }\right]=\left[\frac{2\pi x}{\lambda t}\right]$.
In the option (d), $\frac{x}{\lambda }$ is unitless. This is not the case with $\frac{c}{\lambda }$.