Question

The electric field of a plane electromagnetic wave is given by $\vec y = {E_0}\hat i\cos (kz)(\omega t)$. The corresponding magnetic field \[\vec B\] is then given by:
A.)$\vec B = \dfrac{{{E_0}}}{c}\hat i\sin (kz)\cos (\omega t)$
B.)\[\vec B = \dfrac{{{E_0}}}{c}\hat j\sin (kz)\cos (\omega t)\]
C.)\[\vec B = \dfrac{{{E_0}}}{c}\hat k\sin (kz)\cos (\omega t)\]
D.)\[\vec B = \dfrac{{{E_0}}}{c}\hat j\cos (kz)\sin (\omega t)\]

Answer 1

Hint – You can start by defining what an electromagnetic wave is and what its properties are. Then use the Maxwell equation for electromagnetic wave, i.e. \[\vec \nabla \times \vec E = - \dfrac{{\partial B}}{{\partial t}}\] to calculate the magnetic field (\[\vec B\]).

Complete step-by-step answer:
In the problem given we are given the following equation
$\vec y = {E_0}\hat i\cos (kz)(\omega t)$
As you can see that the wave is travelling in the z-axis and the vibrations of the electric field (\[\vec E\]) are along the positive x-axis. Hence the vibrations in the magnetic field (\[\vec B\]) will be along the y-axis.
Using Maxwell’s equation, we know

\[\vec \nabla \times \vec E = - \dfrac{{\partial B}}{{\partial t}}\]
\[\dfrac{{\partial E}}{{\partial Z}} = - \dfrac{{\partial B}}{{\partial t}}\]
And \[{B_0} = \dfrac{{{E_0}}}{c}\]
So, \[\vec B = \dfrac{{{{\vec E}_0}}}{c}\hat j\sin (kz)\cos (wt)\]

Additional Information:
Electromagnetic waves also known as EM waves are waves that have electric and magnetic vibrations perpendicular and perpendicular to the direction of motion of the electromagnetic wave. Charged particles undergoing acceleration release energy in the form of electromagnetic waves. Electromagnetic waves have energy, momentum and also angular momentum and carry them away from the source. Electromagnetic waves transfer these quantities to whatever target they might hit. A unique feature of electromagnetic waves is that it does not require medium to travel unlike sound waves that are a longitudinal wave that require a medium. They are also much faster. Electromagnetic waves are the fastest waves ever. Light, the fastest entity known, is an example of electromagnetic waves.

Note – Electromagnetic waves are very important for us. From the light rays from the sun, that the plants utilize to photosynthesize and make food (the primary source of energy for all organisms for earth). We also have UV light for sterilization and radio waves for communication that is widely used even now.