Q.The energies $E_1$ and $E_2$ of two radiations are 25 eV and 50 eV respectively. The relation between their wavelengths i.e. $\lambda_1$ and $\lambda_2$ will be

Niels Bohr rescued Rutherford's nuclear atom from classical electromagnetism's prediction of collapse by proposing that electrons occupy only certain fixed, non-radiating stationary orbits, in which the angular momentum is quantised as an integer multiple of $h/2\pi$: $mvr=nh/2\pi$. While an electron sits in a stationary orbit it does not radiate energy, no matter what classical theory says about accelerated charges - this single postulate is what makes atoms stable. An electron only emits or absorbs energy when it jumps between orbits, and the photon involved carries exactly the energy difference between the two orbits, $E_2-E_1=h\nu$.

For any one-electron ("hydrogen-like") species of atomic number $Z$ - hydrogen itself, or an ion such as $\text{He}^+$ or $\text{Li}^{2+}$ - solving the model gives closed-form results for the radius and energy of the $n$th orbit:

The negative sign reflects that a bound electron is at lower energy than a free electron at infinite separation (taken as zero). Because energy depends on $1/n^2$, energy gaps between successive levels shrink rapidly as $n$ grows - so transitions between high, closely-spaced levels release far less energetic photons than transitions down to the ground state. Since $E_n\propto Z^2$, one-electron ions of different $Z$ (like $\text{H}$ and $\text{He}^+$) have spectra of an identical mathematical shape, just scaled by $Z^2$ - hydrogen-like species with the same electron count really do behave like scaled copies of each other.

Despite its success with hydrogen, the model applies only to one-electron species, cannot account for spectral-line splitting in magnetic or electric fields (the Zeeman and Stark effects), and offers no physical reason for its own central assumption of quantised angular momentum - a gap only closed once de Broglie's wave picture of the electron was developed.