Answer
Verified
419.1k+ views
Hint: Wave function for a given plane electromagnetic wave is written as
$ \text{B=Bo}\sin \left( kx-wt \right) $
In case of plane electromagnetic waves, direction of propagation is along the cross product $ \overrightarrow{E}\times \overrightarrow{B} $ . Hence, if direction of propagation is along x axis then $ \overrightarrow{E} $ is along y-axis and $ \overrightarrow{B} $ is along Z axis. From this, the direction of the magnetic field is found.
Complete step by step solution
We know that
$ \text{Bo}=\dfrac{\text{Eo}}{\text{c}} $
Now,
$ \text{Eo=27} $ …. Given
$ \text{c}=3\times {{10}^{8}} $
Hence
$ \text{Bo}=\dfrac{27}{3\times {{10}^{8}}}=9\times {{10}^{-8}} $
As the direction of propagation is along the x-axis, therefore $ \text{Bo} $ is along the z axis.
So $ \overrightarrow{\text{Bo}}=\left( 9\times {{10}^{-8}} \right)\hat{k} $
Now $ k=\dfrac{2\pi }{\lambda } $ and $ w=\dfrac{2\pi }{T} $
So putting these values in equation
$ \text{B=Bo }\sin \left( kx-wt \right) $
Now
$ \begin{align}
& \lambda =\dfrac{c}{v} \\
& =\dfrac{3\times {{10}^{8}}}{2\times {{10}^{14}}}=1\cdot 5\times {{10}^{-6}}m \\
\end{align} $
And
$ T=\dfrac{1}{v}=\dfrac{1}{2\times {{10}^{14}}}=0\cdot 5\times {{10}^{-14}}\text{ sec} $
Putting these values in equation (1), we get
$ \begin{align}
& \text{B}=\left( 9\times {{10}^{-8}} \right)\hat{k}\text{ sin}\left[ 2\pi \left( \dfrac{x}{1\cdot 5\times {{10}^{-6}}}-\dfrac{t}{0\cdot 5\times {{10}^{14}}} \right) \right] \\
& \text{B=}\left( 9\times {{10}^{-8}} \right)\hat{k}\text{ }\sin \left[ 2\pi \left( 0\cdot 666\times {{10}^{6}}x-2\times {{10}^{14}}t \right) \right] \\
\end{align} $ .
Note
$ E_o $ and $ B_o $ are the maximum magnitude of electric and magnetic fields in case of an electromagnetic wave.
Number of cycles of electric field or magnetic field completed in one second is called frequency of electromagnetic wave.
For mathematical representation, we define angular frequency by multiplying frequency with $ 2\pi $
$ w=2\pi v $
Time period of electromagnetic oscillation can be written as:
$ T=\dfrac{1}{v} $
Distance travelled by wave in one time period is called wavelength and is represented by $ \lambda $
$ \begin{align}
& \lambda =CT \\
& \lambda =\dfrac{C}{V} \\
\end{align} $ .
$ \text{B=Bo}\sin \left( kx-wt \right) $
In case of plane electromagnetic waves, direction of propagation is along the cross product $ \overrightarrow{E}\times \overrightarrow{B} $ . Hence, if direction of propagation is along x axis then $ \overrightarrow{E} $ is along y-axis and $ \overrightarrow{B} $ is along Z axis. From this, the direction of the magnetic field is found.
Complete step by step solution
We know that
$ \text{Bo}=\dfrac{\text{Eo}}{\text{c}} $
Now,
$ \text{Eo=27} $ …. Given
$ \text{c}=3\times {{10}^{8}} $
Hence
$ \text{Bo}=\dfrac{27}{3\times {{10}^{8}}}=9\times {{10}^{-8}} $
As the direction of propagation is along the x-axis, therefore $ \text{Bo} $ is along the z axis.
So $ \overrightarrow{\text{Bo}}=\left( 9\times {{10}^{-8}} \right)\hat{k} $
Now $ k=\dfrac{2\pi }{\lambda } $ and $ w=\dfrac{2\pi }{T} $
So putting these values in equation
$ \text{B=Bo }\sin \left( kx-wt \right) $
Now
$ \begin{align}
& \lambda =\dfrac{c}{v} \\
& =\dfrac{3\times {{10}^{8}}}{2\times {{10}^{14}}}=1\cdot 5\times {{10}^{-6}}m \\
\end{align} $
And
$ T=\dfrac{1}{v}=\dfrac{1}{2\times {{10}^{14}}}=0\cdot 5\times {{10}^{-14}}\text{ sec} $
Putting these values in equation (1), we get
$ \begin{align}
& \text{B}=\left( 9\times {{10}^{-8}} \right)\hat{k}\text{ sin}\left[ 2\pi \left( \dfrac{x}{1\cdot 5\times {{10}^{-6}}}-\dfrac{t}{0\cdot 5\times {{10}^{14}}} \right) \right] \\
& \text{B=}\left( 9\times {{10}^{-8}} \right)\hat{k}\text{ }\sin \left[ 2\pi \left( 0\cdot 666\times {{10}^{6}}x-2\times {{10}^{14}}t \right) \right] \\
\end{align} $ .
Note
$ E_o $ and $ B_o $ are the maximum magnitude of electric and magnetic fields in case of an electromagnetic wave.
Number of cycles of electric field or magnetic field completed in one second is called frequency of electromagnetic wave.
For mathematical representation, we define angular frequency by multiplying frequency with $ 2\pi $
$ w=2\pi v $
Time period of electromagnetic oscillation can be written as:
$ T=\dfrac{1}{v} $
Distance travelled by wave in one time period is called wavelength and is represented by $ \lambda $
$ \begin{align}
& \lambda =CT \\
& \lambda =\dfrac{C}{V} \\
\end{align} $ .
Recently Updated Pages
Who among the following was the religious guru of class 7 social science CBSE
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Trending doubts
Write the difference between order and molecularity class 11 maths CBSE
A rainbow has circular shape because A The earth is class 11 physics CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What are noble gases Why are they also called inert class 11 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Differentiate between calcination and roasting class 11 chemistry CBSE