Question

Given that \[y = A{\text{ }}\sin \left[ {\left( {\dfrac{{2\pi }}{\lambda }\left( {ct{\text{ }} - {\text{ }}x} \right)} \right)} \right]\], where \[{\text{y}}\]and \[{\text{x}}\]are measured in meters. Which of the following statements is true ?
A. The unit of ${\lambda ^{ - 1}}$ is the same as that of $\dfrac{{2\pi }}{\lambda }\:$.
B. The unit of $\lambda $ is the same as that of ${\text{x}}$ but not of $A\:$.
C. The unit of $c$ is the same as that of $\dfrac{{2\pi }}{\lambda }\:$.
D. The unit of $(ct{\text{ }} - {\text{ }}x)$ is the same as that of $\dfrac{{2\pi }}{\lambda }\:$.

Answer 1

Hint: We will perform unit analysis of all the parameters involved in the given expression. Then, we will check each given option one by one to come up to a conclusion. Finally, we will tally our conclusions with the given options and pick a suitable one.

Complete answer:
Firstly, let us check the units of each parameter involved in the given expression.The first involved parameter is $y$. Clearly, is referring to some sort of distance or length. Thus, its unit will be $m$. Again, $A$ also refers to displacement or length. Thus, its unit will also be $m$. Then again,$\lambda $ is wavelength, thus its unit is also $m$. Then, $c$ is the speed of light. Thus, it has a unit of $m{s^{ - 1}}$. Then comes, $t$ referring to time thus having a unit of $s$. Finally there comes $x$ which is a distance or length and thus having a unit of $m$.

Now, according to the first option, we will need to take into consideration the units of ${\lambda ^{ - 1}}$ and $\dfrac{{2\pi }}{\lambda }$. We know, $\lambda $ has unit ${\text{m}}$ and thus we can say that ${\lambda ^{ - 1}}$ has a unit of ${m^{ - 1}}$. Now, $2\pi $ is a constant and thus has no units. Thus, the unit of $\dfrac{{2\pi }}{\lambda }$ will be same as the unit of ${\lambda ^{ - 1}}$ which is ${m^{ - 1}}$. Thus, the statement of the option is a correct one.

The statement of the second option is clearly not true as $\lambda $, $A$ and $x$ has the unit of length that is $m$. Also, the third statement is also a vague one as $c$ is referring to the unit of speed but $\dfrac{{2\pi }}{\lambda }$ refers to the unit of length inverted which is clearly not the same.Finally, in the fourth option, the unit of $(ct{\text{ }} - {\text{ }}x)$ will be $(m{s^{ - 1}} \times {\text{ }}s{\text{ }} - {\text{ }}m)$ and thus having a unit of ${\text{m}}$ but the unit of $\dfrac{{2\pi }}{\lambda }$ is ${m^{ - 1}}$.

Thus, the correct option is A.

Note:We are comparing the parameters in S.I. units for a consistent and fair comparison. The students might also do the whole comparison in other unit systems such as CGS or MKS systems. Students should be cautious that all the units used should be of the same unit system for a fair comparison and a reasonable answering method.