$\sqrt{d_{1}}$ and $\sqrt{d_{2}}$ are the impact parameters corresponding to scattering angles $60^{\circ}$ and $90^{\circ}$ respectively, when an $\alpha$ particle is approaching a gold nucleus. For $d_{1}=x d_{2}$, the value of $x$ will be ________

The following diagram indicates the energy levels of a certain atom when the system moves from $2E$ level to $E$, emitting a photon of wavelength $\lambda $. The wavelength of photon produced during its transition from $\frac{4E}{3}$ level to $E$ is

State the limitations of plum pudding model of the atom. 

Given below are two statements:

Statement $I$: Atoms are electrically neutral as they contain equal number of positive and negative charges.

Statement $II$: Atoms of each element are stable and emit their characteristic spectrum.

In the light of the above statements, choose the most appropriate answer from the options given below.

When an $\alpha -$particle of mass $m$ moving with velocity $v$ bombards on a heavy nucleus of charge $Z_e$ , its distance of closest approach from the nucleus depends on $m$ as

What is shown by Thomson's experiments of electric discharge through gases ? And explain the plum pudding model.