Question 7

(a) Using the Bohr’s model calculate the speed of the electron in a hydrogen atom in the n = 1, 2, and 3 levels.

(b) Calculate the orbital period in each of these levels.

Answer

(a) Let ν_{1} be the orbital speed of the electron in a hydrogen atom in the ground state level, n_{1} = 1. For charge (e) of an electron, ν_{1} is given by the relation,

ν_{ 1} = e^{2}/n_{1}4πϵ_{0}(h/2π) = e^{2}/2ϵ_{0}h

Where, e = 1.6 × 10^{−19} C

ϵ_{0} = Permittivity of free space = 8.85 × 10^{-12} N^{−1} C^{2} m^{−2}

h = Planck’s constant = 6.62 × 10^{−34} Js

∴ ν_{1} = (1.6x10^{-19})2/2x8.85x10^{-12}x6.62x10^{-34} = 0.0218 x 10^{8} = 2.18 x 10^{6} *m/s*

For level n_{2} = 2, we can write the relation for the corresponding orbital speed as:

ν_{2} = e^{2}/n_{2}2ϵ_{0}h = (1.6x10^{-19})2/2x2x8.85x10^{-12}x6.62x10^{-34} = 1.09 x 10^{6} m/s

And, for n_{3} = 3, we can write the relation for the corresponding orbital speed as:

ν_{3} = e^{2}/n_{3}2ϵ_{0}h = (1.6x10^{-19})2/3x2x8.85x10^{-12}x6.62x10^{-34} = 7.27 x 10^{5} m/s

Hence, the speed of the electron in a hydrogen atom in n = 1, n=2, and n=3 is 2.18 × 10 ^{6} m/s, 1.09 × 10 ^{6} m/s, 7.27 × 10 ^{5} m/s respectively.

(b) Let T _{1} be the orbital period of the electron when it is in level n_{1} = 1.

Orbital period is related to orbital speed as:

T_{1} = 2πr_{1}/ν_{ 1}

Where, r_{1} = Radius of the orbit

= n_{1}^{2}h^{2}ϵ_{0}/πme^{2}

h = Planck’s constant = 6.62 × 10^{−34} Js

e = Charge on an electron = 1.6 × 10^{−19} C

ϵ_{0} = Permittivity of free space = 8.85 × 10^{−12} N^{−1} C^{2} m^{−2}

m = Mass of an electron = 9.1 × 10^{−31} kg

∴ T_{1} = 2πr_{1}/ν _{1}

= (2πx(1)^{2}x(6.62x10^{-34})2x8.85x10^{-12})/2.18x10^{6}xπx9.1x10^{-31}x(1.6x10^{-19})^{2}

= 15.27x10^{-17} = 1.527x10^{-16} s

For level n _{2} = 2, we can write the period as:

T_{2} = 2πr_{2}/ν _{2}

Where, r_{2} = Radius of the electron in n_{2} = 2

= (n_{2})^{2}h^{2}ϵ_{0}/πme^{2}

∴ T_{2} = 2πr_{2}/ν_{2}

= (2πx(2)^{2}x(6.62x10^{-34})^{2}x8.85x10^{-12})/1.09 x 10^{6} x π x 9.1 x 10^{-31} x (1.6 x 10^{-19})^{2}

= 1.22 x 10^{-15} s

And, for level n_{ 3} = 3, we can write the period as:

T_{3} = 2πr_{3}/ν _{3 }

Where, r _{3} = Radius of the electron in n_{ 3} = 3

= (n_{3})^{2}h^{2}ϵ_{0}/πme^{2}

∴ T_{3} = 2πr_{3}/ν _{3}

= (2πx(3)^{2}x(6.62x10^{-34})2x8.85x10^{-12})/7.27 x 10^{5} x π x 9.1 x 10^{-31} x (1.6 x 10^{-19})^{2}

= 4.12 x 10^{-15} s

Hence, the orbital period in each of these levels is 1.52 × 10 ^{−16} s, 1.22 × 10 ^{−15} s, and 4.12 × 10 ^{−15} s respectively.

- Q:-
An infinite line charge produces a field of 9 × 10

^{4}N/C at a distance of 2 cm. Calculate the linear charge density. - Q:-
A polythene piece rubbed with wool is found to have a negative charge of 3 × 10

^{−7}C.(a) Estimate the number of electrons transferred (from which to which?)

(b) Is there a transfer of mass from wool to polythene?

- Q:- ">
A point charge +10 μC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)

- Q:-
What is the force between two small charged spheres having charges of 2 x 10

^{-7}C and 3 x 10^{-7}C placed 30 cm apart in air? - Q:-
Consider a uniform electric field E = 3 × 10

^{3}îN/C.(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane?

(b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?

- Q:-
Two point charges q

_{A}= 3 μC and q_{B}= −3 μC are located 20 cm apart in vacuum.(a) What is the electric field at the midpoint O of the line AB joining the two charges?

(b) If a negative test charge of magnitude 1.5 × 10

^{−9}C is placed at this point, what is the force experienced by the test charge? - Q:-
A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 10

^{3}N/C and points radially inward, what is the net charge on the sphere? - Q:-
A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 μC/m

^{2}.(a) Find the charge on the sphere.

(b) What is the total electric flux leaving the surface of the sphere?

- Q:-
Two charges 5 x 10

^{-8}C and -3 x 10^{-8}C are located 16 cm apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero. - Q:-
A 600 pF capacitor is charged by a 200 V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. How much electrostatic energy is lost in the process?

- Q:-
A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 10

^{3}N/C and points radially inward, what is the net charge on the sphere? - Q:- The electrostatic force on a small sphere of charge 0.4 μC due to another small sphere of charge − 0.8 μC in air is 0.2 N.
(a) What is the distance between the two spheres?

(b) What is the force on the second sphere due to the first?

">The electrostatic force on a small sphere of charge 0.4 μC due to another small sphere of charge − 0.8 μC in air is 0.2 N.

(a) What is the distance between the two spheres?

(b) What is the force on the second sphere due to the first?

- Q:-
An infinite line charge produces a field of 9 × 10

^{4}N/C at a distance of 2 cm. Calculate the linear charge density. - Q:- The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 x 10
^{28}m^{-3}. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 x 10^{-6}m^{2}and it is carrying a current of 3.0 A.">The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 x 10

^{28}m^{-3}. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 x 10^{-6}m^{2}and it is carrying a current of 3.0 A. - Q:- Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?">
Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?

- Q:-
(a) In a metre bridge [Fig. 3.27], the balance point is found to be at 39.5 cm from the end

*A*, when the resistor*Y*is of 12.5**Ω**. Determine the resistance of*X*. Why are the connections between resistors in a Wheatstone or meter bridge made of thick copper strips?(b) Determine the balance point of the bridge above if

*X*and*Y*are interchanged.(c) What happens if the galvanometer and cell are interchanged at the balance point of the bridge? Would the galvanometer show any current?

- Q:- ">
A point charge +10 μC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)

- Q:- A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the centre of the coil?
- Q:-
Consider a uniform electric field E = 3 × 10

^{3}îN/C.(a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane?

(b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?

- Q:-
A polythene piece rubbed with wool is found to have a negative charge of 3 × 10

^{−7}C.(a) Estimate the number of electrons transferred (from which to which?)

(b) Is there a transfer of mass from wool to polythene?

- NCERT Chapter

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