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Question: The wavelength of a wave is \(5000\mathop A\limits^0 .\) then the wave number is A. \(2 \times {1...

The wavelength of a wave is 5000A0.5000\mathop A\limits^0 . then the wave number is
A. 2×106m−12 \times {10^6}{m^{ - 1}}
B. 2×10−6m−12 \times {10^{ - 6}}{m^{ - 1}}
C. 5×107m−15 \times {10^7}{m^{ - 1}}
D. 5×10−7m−15 \times {10^{ - 7}}{m^{ - 1}}

Explanation

Solution

As we know that, a wave is a disturbance in a medium which travels from one position to another position in space and with this disturbance energy also travel from one point to another, The energy can be either sound energy waves, light waves or any electromagnetic waves and the distance between two successive crest or trough is known as wavelength of the wave. We will use the basic relation between wavenumber and wavelength in order to calculate the wavenumber of a given wave.

Formula used:
Wavenumber and wavelength are related as,
\mathop \upsilon \limits^\\_ = \dfrac{1}{\lambda }
where λ\lambda is the wavelength of the wave and \mathop \upsilon \limits^\\_ is the wave number.

Complete step by step answer:
According to the definition of wave number, when we calculate the number of waves in a unit of length, that number of waves per unit length is called wave number and mathematically it is simply the unit fraction of wavelength of the wave. We have given in the question that,
λ=5000Ao=5×10−7m\lambda = 5000\mathop A\limits^o = 5 \times {10^{ - 7}}m
Now, using the formula of wave number, we have
\mathop \upsilon \limits^\\_ = \dfrac{1}{\lambda }
\Rightarrow \mathop \upsilon \limits^\\_ = \dfrac{{{{10}^7}}}{5}
\therefore \mathop \upsilon \limits^\\_ = 2.0 \times {10^6}{m^{ - 1}}
So, the wavenumber for a given wavelength is \mathop \upsilon \limits^\\_ = 2.0 \times {10^6}{m^{ - 1}}.

Hence, the correct option is A.

Note: It should be noted that, the basic unit of conversion is used while solving the question is 1Ao=10−10m1\mathop A\limits^o = {10^{ - 10}}m and when we have the general expression of a travelling electromagnetic waves in sine and cosine functions then the relation between wavelength and wavenumber is written as k=2πλk = \dfrac{{2\pi }}{\lambda } and it has a direction of propagation of wave and this vector k⃗\vec k when represented in vector form is known as wave vector.