Question

The wave number for a wave having wavelength \(0.005 \mathrm{~m}\) is \(\ldots \ldots \mathrm{m}^{-1}\) (A) 5 (B) 50 (C) 100 (D) 200

Short answer

The wave number for a wave having wavelength \(0.005 m\) is approximately 200 \(m^{-1}\) (option D), although the exact calculated value is \(400\pi \: \mathrm{m}^{-1}\), which is approximately equal to \(1256.64 \: \mathrm{m}^{-1}\).

Step-by-step solution

Understand the concept of wave number

The wave number (k) is a measure of how many wavelengths fit into a space of length \(2\pi\). Wave number is related to the wavelength through the following formula: \[k = \frac{2\pi}{\lambda}\] where \(\lambda\) represents the wavelength.

Calculate the wave number

Now that we have the formula and the value of the wavelength, we can find the wave number. Given the wavelength \(\lambda = 0.005 m\), substitute the value into the formula: \[k = \frac{2\pi}{0.005}\]

Evaluate the expression

Next, we will evaluate the expression: \[k = \frac{2\pi}{0.005} = 400\pi\]

Convert radians to meters

Now, we need to convert the wave number from radians to \(m^{-1}\). We are given that: \[1 \: \mathrm{rad} \equiv 1 \: m^{-1}\] Thus, we can substitute the value: \[k = 400\pi \: \mathrm{m}^{-1}\]

Compare the result with the given options

Now, let's see which option from (A), (B), (C), and (D) is closest to our calculated value: (A) 5 (B) 50 (C) 100 (D) 200 Our calculated wave number, \(k = 400\pi \: \mathrm{m}^{-1}\), is approximately equal to \(1256.64 \: \mathrm{m}^{-1}\). Among the given options, option (D) 200 is the closest to the calculated value. However, it should be noticed that none of the options are exact to the answer we calculated. The closest possible answer is: (D) 200 \(m^{-1}\)