Important Formulas: Modern Physics I & II

# Important Modern Physics I & II Formulas for JEE and NEET

``` Page 1

Modern Physics – I
Important Formulae
1.  Bohr's Theory
(i) Bohr's theory is applicable for hydrogen and hydrogen like atoms/ions. For such type
of atoms/ions number of electron is one. Although atomic numbers may be different,
eg,  For
1
H
1
, atomic number Z = 1
For He
+
, atomic number Z =2 and
For Li
+2
, atomic number Z = 3 .
But for all three number of electron is one.
(ii) In nth orbit

2
n 2
0
mv 1 (e)(Ze) nh
. and L mvr
t 4 r 2
? ? ?
? ? ?

After solving these two equations we will get following results
(a)
2
n1
r and r
Zm
??
(b)
0
Z
v and v m
n
??
(c)
2
2
Z
E and E m
n
??
(d)
H
1
r 0.529 ? Å

(e)
H6
1
c
v 2.2 10 m / s
137
? ? ?
(f)
H
1
E 13.6eV ??
(g) K = |E| and U = 2E
(iii) Energy of a photon
hc
E ?
?
.
After substituting values of h and c, we get

o
12375
E(in eV)
(in A)
?
?

(iv) Hydrogen spectrum

Page 2

Modern Physics – I
Important Formulae
1.  Bohr's Theory
(i) Bohr's theory is applicable for hydrogen and hydrogen like atoms/ions. For such type
of atoms/ions number of electron is one. Although atomic numbers may be different,
eg,  For
1
H
1
, atomic number Z = 1
For He
+
, atomic number Z =2 and
For Li
+2
, atomic number Z = 3 .
But for all three number of electron is one.
(ii) In nth orbit

2
n 2
0
mv 1 (e)(Ze) nh
. and L mvr
t 4 r 2
? ? ?
? ? ?

After solving these two equations we will get following results
(a)
2
n1
r and r
Zm
??
(b)
0
Z
v and v m
n
??
(c)
2
2
Z
E and E m
n
??
(d)
H
1
r 0.529 ? Å

(e)
H6
1
c
v 2.2 10 m / s
137
? ? ?
(f)
H
1
E 13.6eV ??
(g) K = |E| and U = 2E
(iii) Energy of a photon
hc
E ?
?
.
After substituting values of h and c, we get

o
12375
E(in eV)
(in A)
?
?

(iv) Hydrogen spectrum

(v) In first orbit of hydrogen atom,
E = -13.6eV, K = 13.6eV and U=-272eV
Similarly, in second orbit,
E = -3.4 eV, K = 3.4 eV and U = -6.8 eV
(vi) n = 1 is ground state, n = 2 is first excited state and n = 3 is second excited state.
(vii) If an electron jumps from n
1
state to n
2
(< n
1
) state, then wavelength of photon
emitted in this process will be given by,

12
nn
hc
EE ??
?

or
12
o
nn
12375
(in A)
(E E )in eV
??
?

Here,
1
2
n 2
1
13.6 Z
E eV
n
??
?
and
2
2
n 2
2
13.6 Z
E eV
n
??
?
(viii) Rhc = 1 Rydberg = 13.6 eV
(ix) R = Rydberg constant = 1.09 × 10
7
m
-1

(x) As n increases
(a) Angular momentum, time period, potential energy and total energy will
increase.
(b) Speed, kinetic energy and frequency will decrease.
(xi) Total number of emission lines from nth state to ground state are,
n(n 1)
2
?
.
2.  Matter Wave or de-Broglie Wave
(i) Every matter particle having some linear momentum is associated with a wave called
matter wave or de-Broglie wave.
(ii) Wavelength of this wave is given by

h h h h
mv 2Km 2qVm
? ? ? ? ?
?

(iii) For an electron,
o
150
(in A)
V(in volt)
??
3.  Electromagnetic Waves
(i) EM-waves have dual nature. Particle as well as a wave.
(ii) Interference, diffraction or polarization can be explained by wave nature of EM
waves. Photoelectric effect, Compton effect etc is explained by particle nature of EM
waves.
Page 3

Modern Physics – I
Important Formulae
1.  Bohr's Theory
(i) Bohr's theory is applicable for hydrogen and hydrogen like atoms/ions. For such type
of atoms/ions number of electron is one. Although atomic numbers may be different,
eg,  For
1
H
1
, atomic number Z = 1
For He
+
, atomic number Z =2 and
For Li
+2
, atomic number Z = 3 .
But for all three number of electron is one.
(ii) In nth orbit

2
n 2
0
mv 1 (e)(Ze) nh
. and L mvr
t 4 r 2
? ? ?
? ? ?

After solving these two equations we will get following results
(a)
2
n1
r and r
Zm
??
(b)
0
Z
v and v m
n
??
(c)
2
2
Z
E and E m
n
??
(d)
H
1
r 0.529 ? Å

(e)
H6
1
c
v 2.2 10 m / s
137
? ? ?
(f)
H
1
E 13.6eV ??
(g) K = |E| and U = 2E
(iii) Energy of a photon
hc
E ?
?
.
After substituting values of h and c, we get

o
12375
E(in eV)
(in A)
?
?

(iv) Hydrogen spectrum

(v) In first orbit of hydrogen atom,
E = -13.6eV, K = 13.6eV and U=-272eV
Similarly, in second orbit,
E = -3.4 eV, K = 3.4 eV and U = -6.8 eV
(vi) n = 1 is ground state, n = 2 is first excited state and n = 3 is second excited state.
(vii) If an electron jumps from n
1
state to n
2
(< n
1
) state, then wavelength of photon
emitted in this process will be given by,

12
nn
hc
EE ??
?

or
12
o
nn
12375
(in A)
(E E )in eV
??
?

Here,
1
2
n 2
1
13.6 Z
E eV
n
??
?
and
2
2
n 2
2
13.6 Z
E eV
n
??
?
(viii) Rhc = 1 Rydberg = 13.6 eV
(ix) R = Rydberg constant = 1.09 × 10
7
m
-1

(x) As n increases
(a) Angular momentum, time period, potential energy and total energy will
increase.
(b) Speed, kinetic energy and frequency will decrease.
(xi) Total number of emission lines from nth state to ground state are,
n(n 1)
2
?
.
2.  Matter Wave or de-Broglie Wave
(i) Every matter particle having some linear momentum is associated with a wave called
matter wave or de-Broglie wave.
(ii) Wavelength of this wave is given by

h h h h
mv 2Km 2qVm
? ? ? ? ?
?

(iii) For an electron,
o
150
(in A)
V(in volt)
??
3.  Electromagnetic Waves
(i) EM-waves have dual nature. Particle as well as a wave.
(ii) Interference, diffraction or polarization can be explained by wave nature of EM
waves. Photoelectric effect, Compton effect etc is explained by particle nature of EM
waves.
(iii) E,B and c
??
?
are mutually perpendicular.
(iv) E B c ??
??
?

(v) E and B oscillate in same phase.
eg,   B
x
= B
0
sin( ?t

- ky) then
E
Z
= E
0
sin ( ?t

- ky)
(vi)
12
2 f ,T
f
?
? ? ? ? ?
?

(vii)
0
0
E
c
B
?
(viii) Properties of a photon
(a) E = hf =
hc
?

(b)E (in eV) =
o
12375
(in A) ?

(c) Momentum of photon
Eh
p
c
??
?

(d) Dynamic mass,
2
E
m
c
?
(ix) ?-rays, X-rays, UV rays, visible light, infrared rays, micro waves, radio waves.
In moving from left to right, energy and frequency of wave decrease but wavelength
increases.
(i) These are electromagnetic waves of high energy, high frequency and low wavelength.
(ii) Wavelength of X-rays lies in the range of 1Å to 100 Å.
(iii) Cut off wavelength of X-ray spectrum is given by

o
min
12375
(in A)
V(in volt)
??
(iv) Moseley's law for X-ray spectrum is
v (z b) ??
Here, the constant b depends on the series
b = 1 for K-series and b = 7.4 f or L-series
(v)
Page 4

Modern Physics – I
Important Formulae
1.  Bohr's Theory
(i) Bohr's theory is applicable for hydrogen and hydrogen like atoms/ions. For such type
of atoms/ions number of electron is one. Although atomic numbers may be different,
eg,  For
1
H
1
, atomic number Z = 1
For He
+
, atomic number Z =2 and
For Li
+2
, atomic number Z = 3 .
But for all three number of electron is one.
(ii) In nth orbit

2
n 2
0
mv 1 (e)(Ze) nh
. and L mvr
t 4 r 2
? ? ?
? ? ?

After solving these two equations we will get following results
(a)
2
n1
r and r
Zm
??
(b)
0
Z
v and v m
n
??
(c)
2
2
Z
E and E m
n
??
(d)
H
1
r 0.529 ? Å

(e)
H6
1
c
v 2.2 10 m / s
137
? ? ?
(f)
H
1
E 13.6eV ??
(g) K = |E| and U = 2E
(iii) Energy of a photon
hc
E ?
?
.
After substituting values of h and c, we get

o
12375
E(in eV)
(in A)
?
?

(iv) Hydrogen spectrum

(v) In first orbit of hydrogen atom,
E = -13.6eV, K = 13.6eV and U=-272eV
Similarly, in second orbit,
E = -3.4 eV, K = 3.4 eV and U = -6.8 eV
(vi) n = 1 is ground state, n = 2 is first excited state and n = 3 is second excited state.
(vii) If an electron jumps from n
1
state to n
2
(< n
1
) state, then wavelength of photon
emitted in this process will be given by,

12
nn
hc
EE ??
?

or
12
o
nn
12375
(in A)
(E E )in eV
??
?

Here,
1
2
n 2
1
13.6 Z
E eV
n
??
?
and
2
2
n 2
2
13.6 Z
E eV
n
??
?
(viii) Rhc = 1 Rydberg = 13.6 eV
(ix) R = Rydberg constant = 1.09 × 10
7
m
-1

(x) As n increases
(a) Angular momentum, time period, potential energy and total energy will
increase.
(b) Speed, kinetic energy and frequency will decrease.
(xi) Total number of emission lines from nth state to ground state are,
n(n 1)
2
?
.
2.  Matter Wave or de-Broglie Wave
(i) Every matter particle having some linear momentum is associated with a wave called
matter wave or de-Broglie wave.
(ii) Wavelength of this wave is given by

h h h h
mv 2Km 2qVm
? ? ? ? ?
?

(iii) For an electron,
o
150
(in A)
V(in volt)
??
3.  Electromagnetic Waves
(i) EM-waves have dual nature. Particle as well as a wave.
(ii) Interference, diffraction or polarization can be explained by wave nature of EM
waves. Photoelectric effect, Compton effect etc is explained by particle nature of EM
waves.
(iii) E,B and c
??
?
are mutually perpendicular.
(iv) E B c ??
??
?

(v) E and B oscillate in same phase.
eg,   B
x
= B
0
sin( ?t

- ky) then
E
Z
= E
0
sin ( ?t

- ky)
(vi)
12
2 f ,T
f
?
? ? ? ? ?
?

(vii)
0
0
E
c
B
?
(viii) Properties of a photon
(a) E = hf =
hc
?

(b)E (in eV) =
o
12375
(in A) ?

(c) Momentum of photon
Eh
p
c
??
?

(d) Dynamic mass,
2
E
m
c
?
(ix) ?-rays, X-rays, UV rays, visible light, infrared rays, micro waves, radio waves.
In moving from left to right, energy and frequency of wave decrease but wavelength
increases.
(i) These are electromagnetic waves of high energy, high frequency and low wavelength.
(ii) Wavelength of X-rays lies in the range of 1Å to 100 Å.
(iii) Cut off wavelength of X-ray spectrum is given by

o
min
12375
(in A)
V(in volt)
??
(iv) Moseley's law for X-ray spectrum is
v (z b) ??
Here, the constant b depends on the series
b = 1 for K-series and b = 7.4 f or L-series
(v)
(vi)
(vii)
22
12
1 1 1
(Z b)R
nn
??
? ? ?
??
?
??

Here, n
1
< n
2

(viii) Continuous X-Ray spectrum

Here,
o
min
12375
(in A)
V(in volt)
?? = cut-off wavelength.
(ix)
This is actual spectrum of X-rays including continuous and characteristic X-rays.
5.  Photoelectric Effect
(i) K
max
=
2
max
1
mv
2
=E - W = hf - hf
0
= eV
0
=
0
hc hc
?
??

(ii) Graphs K
max
= hf - W

Slope
1
= Slope
2
= Planck's constant f
01
= Threshold frequency of metal-1
f
02
= Threshold frequency of metal-2
W
1
= Work function of metal-1 and
Page 5

Modern Physics – I
Important Formulae
1.  Bohr's Theory
(i) Bohr's theory is applicable for hydrogen and hydrogen like atoms/ions. For such type
of atoms/ions number of electron is one. Although atomic numbers may be different,
eg,  For
1
H
1
, atomic number Z = 1
For He
+
, atomic number Z =2 and
For Li
+2
, atomic number Z = 3 .
But for all three number of electron is one.
(ii) In nth orbit

2
n 2
0
mv 1 (e)(Ze) nh
. and L mvr
t 4 r 2
? ? ?
? ? ?

After solving these two equations we will get following results
(a)
2
n1
r and r
Zm
??
(b)
0
Z
v and v m
n
??
(c)
2
2
Z
E and E m
n
??
(d)
H
1
r 0.529 ? Å

(e)
H6
1
c
v 2.2 10 m / s
137
? ? ?
(f)
H
1
E 13.6eV ??
(g) K = |E| and U = 2E
(iii) Energy of a photon
hc
E ?
?
.
After substituting values of h and c, we get

o
12375
E(in eV)
(in A)
?
?

(iv) Hydrogen spectrum

(v) In first orbit of hydrogen atom,
E = -13.6eV, K = 13.6eV and U=-272eV
Similarly, in second orbit,
E = -3.4 eV, K = 3.4 eV and U = -6.8 eV
(vi) n = 1 is ground state, n = 2 is first excited state and n = 3 is second excited state.
(vii) If an electron jumps from n
1
state to n
2
(< n
1
) state, then wavelength of photon
emitted in this process will be given by,

12
nn
hc
EE ??
?

or
12
o
nn
12375
(in A)
(E E )in eV
??
?

Here,
1
2
n 2
1
13.6 Z
E eV
n
??
?
and
2
2
n 2
2
13.6 Z
E eV
n
??
?
(viii) Rhc = 1 Rydberg = 13.6 eV
(ix) R = Rydberg constant = 1.09 × 10
7
m
-1

(x) As n increases
(a) Angular momentum, time period, potential energy and total energy will
increase.
(b) Speed, kinetic energy and frequency will decrease.
(xi) Total number of emission lines from nth state to ground state are,
n(n 1)
2
?
.
2.  Matter Wave or de-Broglie Wave
(i) Every matter particle having some linear momentum is associated with a wave called
matter wave or de-Broglie wave.
(ii) Wavelength of this wave is given by

h h h h
mv 2Km 2qVm
? ? ? ? ?
?

(iii) For an electron,
o
150
(in A)
V(in volt)
??
3.  Electromagnetic Waves
(i) EM-waves have dual nature. Particle as well as a wave.
(ii) Interference, diffraction or polarization can be explained by wave nature of EM
waves. Photoelectric effect, Compton effect etc is explained by particle nature of EM
waves.
(iii) E,B and c
??
?
are mutually perpendicular.
(iv) E B c ??
??
?

(v) E and B oscillate in same phase.
eg,   B
x
= B
0
sin( ?t

- ky) then
E
Z
= E
0
sin ( ?t

- ky)
(vi)
12
2 f ,T
f
?
? ? ? ? ?
?

(vii)
0
0
E
c
B
?
(viii) Properties of a photon
(a) E = hf =
hc
?

(b)E (in eV) =
o
12375
(in A) ?

(c) Momentum of photon
Eh
p
c
??
?

(d) Dynamic mass,
2
E
m
c
?
(ix) ?-rays, X-rays, UV rays, visible light, infrared rays, micro waves, radio waves.
In moving from left to right, energy and frequency of wave decrease but wavelength
increases.
(i) These are electromagnetic waves of high energy, high frequency and low wavelength.
(ii) Wavelength of X-rays lies in the range of 1Å to 100 Å.
(iii) Cut off wavelength of X-ray spectrum is given by

o
min
12375
(in A)
V(in volt)
??
(iv) Moseley's law for X-ray spectrum is
v (z b) ??
Here, the constant b depends on the series
b = 1 for K-series and b = 7.4 f or L-series
(v)
(vi)
(vii)
22
12
1 1 1
(Z b)R
nn
??
? ? ?
??
?
??

Here, n
1
< n
2

(viii) Continuous X-Ray spectrum

Here,
o
min
12375
(in A)
V(in volt)
?? = cut-off wavelength.
(ix)
This is actual spectrum of X-rays including continuous and characteristic X-rays.
5.  Photoelectric Effect
(i) K
max
=
2
max
1
mv
2
=E - W = hf - hf
0
= eV
0
=
0
hc hc
?
??

(ii) Graphs K
max
= hf - W

Slope
1
= Slope
2
= Planck's constant f
01
= Threshold frequency of metal-1
f
02
= Threshold frequency of metal-2
W
1
= Work function of metal-1 and
W
2
= Work function of metal-2
From the equation K
max
=hf -W, we can see that K
max
versus f graph is a straight line with
positive slope h (a universal constant) and negative intercept W (varies metal to metal).

Slope
1
= Slope
2
=
h
e
= a universal constant
From the equation,
0
hW
Vf
ee
??
??
??
??
,

we can see that stopping potential V
0
versus
frequency f graph is a straight line with positive slope
h
e
and negative slope
W
e
.
(iii) Threshold wavelength
0
hc
W
??
(iv) Threshold frequency
0
W
f
h
?
(v) Photoelectric current versus potential graph

Saturation current I
2
depends upon number of photons incident on metallic plate per sec
and stopping potential depends on the frequency of incident photons.
(vi) For photoemission to take place,
E ? W, ? = ?
0

and
(vii) Kinetic energy of photoelectrons varies between 0 and K
max
.
```

## Physics Class 12

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## FAQs on Important Modern Physics I & II Formulas for JEE and NEET

 1. What are the important formulas in Modern Physics I & II?
Ans. The important formulas in Modern Physics I & II include: - Einstein's mass-energy equivalence formula: E = mc^2 - Planck's energy quantization formula: E = hf - de Broglie wavelength formula: λ = h/p - Schrödinger equation: Ĥψ = Eψ - Heisenberg's uncertainty principle: ΔxΔp ≥ h/4π
 2. What is the significance of Einstein's mass-energy equivalence formula (E = mc^2)?
Ans. Einstein's mass-energy equivalence formula states that mass and energy are interchangeable and connected by the speed of light squared. This formula revolutionized our understanding of physics by showing that mass can be converted into energy and vice versa. It laid the foundation for nuclear energy and the development of nuclear power plants.
 3. How is Planck's energy quantization formula (E = hf) relevant in Modern Physics I & II?
Ans. Planck's energy quantization formula states that energy is quantized in discrete units called quanta, where E represents energy, h is Planck's constant, and f is the frequency of the electromagnetic radiation. This formula is essential in understanding the behavior of particles at the atomic and subatomic level, and it forms the basis of quantum mechanics.
 4. Explain the concept of de Broglie wavelength and its significance in Modern Physics I & II.
Ans. The de Broglie wavelength, given by λ = h/p, relates the wavelength of a particle to its momentum, where λ represents the wavelength, h is Planck's constant, and p is the momentum. This concept suggests that particles, such as electrons and protons, exhibit wave-like properties. It is significant in Modern Physics as it helps explain phenomena such as electron diffraction and wave-particle duality.
 5. How does the Heisenberg uncertainty principle impact Modern Physics I & II?
Ans. The Heisenberg uncertainty principle, ΔxΔp ≥ h/4π, states that there is a fundamental limit to the simultaneous measurement of a particle's position (Δx) and momentum (Δp). It implies that the more precisely one measures the position of a particle, the less precisely one can know its momentum, and vice versa. This principle has profound implications in understanding the behavior of particles at the quantum level and is a fundamental concept in quantum mechanics.

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