Orbits of a particle moving in a circle are such that the perimeter...

Orbits of a particle moving in a circle are such that the perimeter of the orbit equals an integer number of de-Broglie wavelengths of the particle. For a charged particle moving in a plane perpendicular to a magnetic field, the radius of the

n^{\text{th }}

orbital will therefore be proportional to :

Options

A $n^2$
B $n$
C $n^{1 / 2}$
D $n^{1 / 4}$

Correct answer

C. $n^{1 / 2}$

Step-by-step solution

According to the question,

\begin{aligned}& 2 \pi \mathrm{r}=\mathrm{n} \lambda=\frac{\mathrm{nh}}{\mathrm{p}}=\frac{\mathrm{nh}}{\mathrm{mv}}\\ & \text{ or } \mathrm{mvr}=\frac{\mathrm{nh}}{2 \pi} \text{ or } \mathrm{mv}=\frac{\mathrm{nh}}{2 \pi \mathrm{r}}\\ & \mathrm{F}=\mathrm{qv}_{\mathrm{B}}=\frac{\mathrm{mv}^2}{\mathrm{r}}\\ & \text{ or, } \mathrm{q}_{\mathrm{B}}=\frac{\mathrm{mv}}{\mathrm{r}}=\frac{\mathrm{nh}}{2 \pi \mathrm{r} \cdot \mathrm{r}}\\ & \text{ or, } \mathrm{r}^2=\frac{\mathrm{nh}}{2 \pi \mathrm{qB}}\\ & \text{ or, } \mathrm{r}=\sqrt{\frac{\mathrm{nh}}{2 \pi \mathrm{qB}}}\\ & \text{ i.e., } \mathrm{r} \propto \mathrmn^{1 / 2} \end{aligned}

Practice Atomic Physics on Quantrex Academy →

More from Atomic Physics

Consider a hydrogen atom with v_k , r_k , and K_k denoting the velocity, orbital radius… JEE Advanced 2026 The ratio of momentum of the photons of the 1^ st and 2^ nd line of Balmer series of… JEE Main 2026 In the hydrogen atom, the electron makes a transition from the higher orbit ( i ) to a… JEE Main 2026 In Rutherford's alpha-particle scattering experiment, only a few alpha particles rebound… JEE Main 2026 Using Bohr's model, calculate the ratio of the magnetic fields generated due to the… JEE Main 2026 Angular momentum of an electron in a hydrogen atom is (3h)/(π) , then the energy of the… JEE Main 2026 Two electrons are moving in orbits of two hydrogen like atoms with speeds 3 × 10^ 5 ~m /… JEE Main 2026 In hydrogen atom spectrum, (R arrow Rydberg's constant ) A. the maximum wavelength of the… JEE Main 2026 Full Atomic Physics list All JEE Main PYQs