Given that $y = A \sin \left( \frac{2\pi}{\lambda} (ct - x) \right)$, where $x$ and $y$ are measured in…
Given that $y = A \sin \left( \frac{2\pi}{\lambda} (ct - x) \right)$, where $x$ and $y$ are measured in metres. Which of the following statement is true? [CG PMT 2015]
Unit of $\lambda$ is same as that of $x$ and $A$
Unit of $\lambda$ is same as that of $x$ and not of $A$
Unit of $c$ is same as that of $\frac{2\pi}{\lambda}$
Unit of $(ct - x)$ is same as that of $\frac{2\pi}{\lambda}$
Solution
Here, $y = A \sin \left(\frac{2\pi}{\lambda} (ct - x)\right)$
where, $x$ and $y$ are measured in metres.
$A$ is amplitude of wave, whose unit is metre and $x$ is also in metre (given).
$\therefore$ Unit of $\lambda$ is same as that of $x$ and $A$, i.e. metre.
However, unit of $c$ (speed of light wave) is $\text{ms}^{-1}$, which is not same as the unit of $2\pi/\lambda$, i.e. $\text{metre}^{-1}$. Similarly, unit of $(ct - x)$ is metre which is not same as the unit of $2\pi/\lambda$.