Given that: $y=A \sin \left[\left(\frac{2 \pi}{\lambda}\right)(c t-x)\right]$, where $y$ and $x$ are…
Given that: $y=A \sin \left[\left(\frac{2 \pi}{\lambda}\right)(c t-x)\right]$, where $y$ and $x$ are measured in the unit of length.
Which of the following statements is true ?
The unit of $\lambda$ is same as that of $x$ and $A$
The unit of $\lambda$ is same as that of $x$ but not of $A$
The unit of $c$ is same as that of $2 \pi / \lambda \mathrm{c}$
The unit of $(c t-x)$ is same as that of $2 \pi / \lambda$
Solution
Unit of ct must be same as unit of \(\mathrm{x}\). Since argument of sin has to be dimensionless, thus dimensions of \(\mathrm{x}\) and ct must be same as dimensions of \(\lambda\). Thus dimension of \(\lambda\) equals 'L'. Also, dimension of y equals 'L' and hence dimension of A equals 'L' because sin term is dimensionless. Thus statement 1 is correct.
All others are incorrect:
Dimension of ct-x is same as \(\lambda\), not \(\frac{1}{\lambda}\). Thus \(\mathrm{D}\) is incorrect. \(\mathrm{C}\) is incorrect because dimension of \(c\) is same as dimension of \(\frac{\lambda}{t}\).
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Asked in: JEE Mains - Units and Dimensions - Test 2