Download App

Atoms — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsAtoms5 Marks
Question
The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about $10^{–40}$. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.
✓

Answer

Radius of the first Bohr orbit is given by the relation,
$\text{r}_1=\frac{4\pi\in_0\Big(\frac{\text{h}}{2\pi}\Big)^2}{\text{m}_\text{e}\text{e}^2}\ \dots(1)$
Where,
$\in_0$ = Permittivity of free space
h = Planck ' s constant $= 6.63 \times 10^{-34}$ Js
me = Mass of an electron $= 9.1 \times 10^{-31}$ kg
e = Charge of an electron $= 1.9 \times 10^{-19}$ C
mp = Mass of a proton $= 1.67 \times 10^{-27}$ kg
r = Distance between the electron and the proton
Coulomb attraction between an electron and a proton is given as:
$\text{F}_\text{C}=\frac{\text{e}^2}{4\pi\in_0\text{r}^2}\ \dots(2)$
Gravitational force of attraction between an electron and a proton is given as:
$\text{F}_\text{G}=\frac{\text{Gm}_\text{p}\text{m}_\text{c}}{\text{r}^2}\ \dots(3)$
Where,
G = Gravitational constant $= 6.67 \times 10^{-11}\ N\ m^2/ kg^2$
If the electrostatic (Coulomb) force and the gravitational force between an electron and a proton are equal, then we can write:
$\therefore\ \text{FG}=\text{FC}$
$\frac{\text{Gm}_\text{p}\text{m}_\text{c}}{\text{r}^2}=\frac{\text{e}^2}{4\pi\in_0\text{r}^2}$
$\therefore\ \frac{\text{e}^2}{4\pi\in_0}=\text{Gm}_\text{p}\text{m}_\text{c}\ \dots(4)$
Putting the value of equation (4) in equation (1), we get:
$\text{r}_1=\frac{\Big(\frac{\text{h}}{2\pi}\Big)^2}{\text{Gm}_\text{p}\text{m}_\text{e}^2}$
$=\frac{\Big(\frac{6.63\times10^{-34}}{2\times3.14}\Big)^2}{6.67\times10^{-11}\times1.67\times10^{-27}\times(9.1\times10^{-31})^2}\approx1.21\times10^{29}\text{ m}$
It is known that the universe is 156 billion light years wide or $1.5 \times 10^{27}\ m$ wide. Hence, we can conclude that the radius of the first Bohr orbit is much greater than the estimated size of the whole universe.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Atoms→Atoms — all question types→Physics STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.
→
Three rigid rods are joined to form an equilateral triangle $\text{ABC}$ of side $1m$. Three particles carrying charges $20\mu\text{C}$ each are attached to the vertices of the triangle. The whole system is at rest in an inertial frame. The resultant force on the charged particle at $A$ has the magnitude:
→
Seven homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced with respect to the one in contact by $\frac{\text{L}}{10}.$ Find the x-coordinate of the centre of mass relative to the origin shown.
→
Consider the situation shown in figure. The elevator is going up with an acceleration of $2.00m/s^2$ and the focal length of the mirror is 12.0cm. All the surfaces are smooth and the pulley is light. The mass-pulley system is released from rest (with respect to the elevator) at t = 0 when the distance of B from the mirror is 42.0cm. Find the distance between the image of the block B and the mirror at t = 0.200s. Take $g = 10m/s^2.$​​​​​​​
→
$4 \times 10^{23}$ tritium atoms are contained in a vessel. The half-life of decay tritium nuclei is 12.3y. Find:
  1. The activity of the sample.
  2. The number of decay in the next 10 hours.
  3. The number of decays in the next 6.15y.
→
Two coherent point sources $S_1$ and $S_2$ vibrating in phase emit light of wavelength $\lambda$. The separation between the sources is $2\lambda$. Consider a line passing through $S_2$ and perpendicular to the line $S_1S_2$. What is the smallest distance from $S_2$ where a minimum of intensity occurs?
→
Two long, straight wires, each carrying a current of 5A, are placed along the X and Y-axes respectively. The currents point along the positive directions of the axes. Find the magnetic fields at the points (a) (1m, 1m), (b) (-1m, 1m), (c) (-1m,-1m) and (d) (1m, -1m),
→
A uniform magnetic field B exists in a cylindrical region, shown dotted in figure. The magnetic field increases at a constant rate $\frac{\text{dB}}{\text{dt}}.$ Consider a circle of radius r coaxial with the cylindrical region.
  1. Find the magnitude of the electric field E at a point on the circumference of the circle.
  2. Consider a point P on the side of the square circumscribing the circle. Show that the component of the induced electric field at P along ba is the same as the magnitude found in part (a).

→
A particle of charge $2.0 \times 10^{-8}C$ and mass $2.0 \times 10^{-10}g$ is projected with a speed of $2.0 \times 10^3m/s^{-1}$ in a region with a uniform magnetic field of 0.10T. The velocity is perpendicular to the field. Find the radius of the circle formed by the particle and also the time period.
→
When a Coolidge tube is operated for some time it becomes hot. Where does the heat come from?
→