In a plane electromagnetic wave,the magnetic field is given by $\vec{B} = 3 \times 10^{-7} \sin (100 \pi x + 10^{12} t) \ T$. The wavelength of the wave is (where $x$ is in meters and $t$ is in seconds). (in $m$)

If ${\varepsilon _0}$ and ${\mu _0}$ are respectively the electric permittivity and the magnetic permeability of free space,and ${\varepsilon}$ and ${\mu}$ are the corresponding quantities in a medium,the refractive index of the medium is:

$A$ plane $EM$ wave travelling in vacuum along $z$-direction is given by $\vec E = E_0 \sin(kz - \omega t) \hat i$ and $\vec B = B_0 \sin(kz - \omega t) \hat j$.
$(i)$ Evaluate $\int \vec E \cdot d\vec l$ over the rectangular loop $1234$ shown in the figure.
$(ii)$ Evaluate $\int \vec B \cdot d\vec s$ over the surface bounded by loop $1234$.
$(iii)$ Use $\int \vec E \cdot d\vec l = -\frac{d\phi_E}{dt}$ to prove $\frac{E_0}{B_0} = c$.
$(iv)$ By using a similar process and the equation $\int \vec B \cdot d\vec l = \mu_0 I + \mu_0 \epsilon_0 \frac{d\phi_E}{dt}$,prove that $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.

The quantities $x = \frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$,$y = \frac{E}{B}$,and $z = \frac{l}{CR}$ are defined,where $C$ is capacitance,$R$ is resistance,$l$ is length,$E$ is electric field,$B$ is magnetic field,and $\varepsilon_{0}, \mu_{0}$ are free space permittivity and permeability respectively. Then:

Answer the following questions:
$(a)$ Long distance radio broadcasts use short-wave bands. Why?
$(b)$ It is necessary to use satellites for long distance $TV$ transmission. Why?
$(c)$ Optical and radio telescopes are built on the ground but $X$-ray astronomy is possible only from satellites orbiting the earth. Why?
$(d)$ The small ozone layer on top of the stratosphere is crucial for human survival. Why?
$(e)$ If the earth did not have an atmosphere,would its average surface temperature be higher or lower than what it is now?
$(f)$ Some scientists have predicted that a global nuclear war on the earth would be followed by a severe 'nuclear winter' with a devastating effect on life on earth. What might be the basis of this prediction?

The transverse nature of electromagnetic waves is demonstrated by which of the following phenomena?