(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.
(a) Let ν1 be the orbital speed of the electron in a hydrogen atom in the ground state level, n1 = 1. For charge (e) of an electron, ν1 is given by the relation,
ν 1 = e2/n14πϵ0(h/2π) = e2/2ϵ0h
Where, e = 1.6 × 10−19 C
ϵ0 = Permittivity of free space = 8.85 × 10-12 N−1 C2 m−2
h = Planck’s constant = 6.62 × 10−34 Js
∴ ν1 = (1.6x10-19)2/2x8.85x10-12x6.62x10-34 = 0.0218 x 108 = 2.18 x 106 m/s
For level n2 = 2, we can write the relation for the corresponding orbital speed as:
ν2 = e2/n22ϵ0h = (1.6x10-19)2/2x2x8.85x10-12x6.62x10-34 = 1.09 x 106 m/s
And, for n3 = 3, we can write the relation for the corresponding orbital speed as:
ν3 = e2/n32ϵ0h = (1.6x10-19)2/3x2x8.85x10-12x6.62x10-34 = 7.27 x 105 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 n1 = 1.
Orbital period is related to orbital speed as:
T1 = 2πr1/ν 1
Where, r1 = Radius of the orbit
= n12h2ϵ0/πme2
h = Planck’s constant = 6.62 × 10−34 Js
e = Charge on an electron = 1.6 × 10−19 C
ϵ0 = Permittivity of free space = 8.85 × 10−12 N−1 C2 m−2
m = Mass of an electron = 9.1 × 10−31 kg
∴ T1 = 2πr1/ν 1
= (2πx(1)2x(6.62x10-34)2x8.85x10-12)/2.18x106xπx9.1x10-31x(1.6x10-19)2
= 15.27x10-17 = 1.527x10-16 s
For level n 2 = 2, we can write the period as:
T2 = 2πr2/ν 2
Where, r2 = Radius of the electron in n2 = 2
= (n2)2h2ϵ0/πme2
∴ T2 = 2πr2/ν2
= (2πx(2)2x(6.62x10-34)2x8.85x10-12)/1.09 x 106 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:
T3 = 2πr3/ν 3
Where, r 3 = Radius of the electron in n 3 = 3
= (n3)2h2ϵ0/πme2
∴ T3 = 2πr3/ν 3
= (2πx(3)2x(6.62x10-34)2x8.85x10-12)/7.27 x 105 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.
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?
An infinite line charge produces a field of 9 × 104 N/C at a distance of 2 cm. Calculate the linear charge density.
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?
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?
A parallel plate capacitor with air between the plates has a capacitance of 8 pF (1pF = 10-12 F). What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant 6?
A regular hexagon of side 10 cm has a charge 5 µC at each of its vertices. Calculate the potential at the centre of the hexagon.
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.)
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 × 103 N/C and points radially inward, what is the net charge on the sphere?
A point charge of 2.0 μC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface?
In an unbiased p-n junction, holes diffuse from the p-region to n-region because
(a) free electrons in the n-region attract them.
(b) they move across the junction by the potential difference.
(c) hole concentration in p-region is more as compared to n-region.
(d) All the above.
Figure shows a potentiometer circuit for comparison of two resistances. The balance point with a standard resistor R = 10.0 Ω is found to be 58.3 cm, while that with the unknown resistance X is 68.5 cm. Determine the value of X. What might you do if you failed to find a balance point with the given cell of emf Ω µ?
A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire?
A circular coil of 16 turns and radius 10 cm carrying a current of 0.75 A rests with its plane normal to an external field of magnitude 5.0 x 10-2 T. The coil is free to turn about an axis in its plane perpendicular to the field direction. When the coil is turned slightly and released, it oscillates about its stable equilibrium with a frequency of 2.0 s-1. What is the moment of inertia of the coil about its axis of rotation?
What is the de Broglie wavelength of
(a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km/s,
(b) a ball of mass 0.060 kg moving at a speed of 1.0 m/s, and
(c) a dust particle of mass 1.0 × 10−9 kg drifting with a speed of 2.2 m/s?
(a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by
Where is a unit vector normal to the surface at a point and σ is the surface charge density at that point. (The direction of is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is σ
(b) Show that the tangential component of electrostatic field is continuous from one side of a charged surface to another.
[Hint: For (a), use Gauss's law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.]
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?
Two tiny spheres carrying charges 1.5 μC and 2.5 μC are located 30 cm apart. Find the potential and electric field:
(a) at the mid-point of the line joining the two charges, and
(b) at a point 10 cm from this midpoint in a plane normal to the line and passing through the mid-point.