Problem set 97

1. In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?
1. Each normal coordinate has an eigen-frequency associated with it
2. The normal coordinates are orthogonal to one another
3. The normal coordinates are all independent
4. The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients

The normal coordinates are orthogonal to one another

Hence, answer is (B)

2. The eigenvalues of a Hermitian matrix are all
1. real
2. imaginary
3. of modulus one
4. real and positive

real

Hence, answer is (A)

3. Given the following table,
   Group-I
   P. Stem-Gerlach experiment
   Q. Zeeman effect
   R. Frank-Hertz experiment
   S. Davisson-Germer experiment
   Group-II
   1. Wave nature of particles
   2. Quantization of energy of electrons in the atoms
   3. Existence of electron spin
   4. Space quantization of angular momentum
   Which one of the following correctly matches the experiments from Group-I to their inferences in Group-II ?
1. P-2, Q-3, R-4, S-1
2. P-1, Q-3, R-2, S-4
3. P-3, Q-4, R-2, S-1
4. P-2, Q-1, R-4, S-3

P. Stem-Gerlach experiment : Existence of electron spin
Q. Zeeman effect:Space quantization of angular momentum
R. Frank-Hertz experiment:Quantization of energy of electrons in the atoms
S. Davisson-Germer experiment:Wave nature of particles


Hence, answer is (C)

4. Which of the following atoms cannot exhibit Bose-Einstein condensation, even in principle?
1. $^{1}H_1$
2. $^{4}H_2$
3. $^{23}Na_{11}$
4. $^{30}K_{19}$
For Bose-Einstein condensation:
Number of electron + number of proton + number of neutron = Even
For $^{30}K_{19}$
Number of proton = 19, Number of electron = 19, Number of neutron = 11.
19 + 19 + 11 = 49 this is odd.
So it will not exhibit Bose-Einstein condensation.

Hence, answer is (D)

5. Two gases separated by an impermeable but movable partition are allowed to freely exchange energy. At equilibrium, the two sides will have the same
1. pressure and temperature.
2. volume and temperature.
3. pressure and volume.
4. volume and energy.

pressure and temperature

Hence, answer is (A)

Problem set 96

1. The transient response of an electronic circuit is tested by giving the following wave form at the input:
1. Sinusoidal
2. Square
3. Triangular
4. Saw-tooth

Square

Hence, answer is (B)

2. Which of the following effects manifests particle nature of light?
1. Photo electric effect
2. Blackbody radiation
3. Interference
4. Diffraction

Photo electric effect

Hence, answer is (A)

3. A ruled grating having 1000 grooves per mm is used for diffraction. If light of wavelength 300 nm is incident at right angle and if $\theta$ is the diffraction angle, $\sin\theta$ for first order diffraction will be equal to:
1. 0.1
2. 0.3
3. 0.2
4. 0.5

Using relation $d\sin\theta=n\lambda$, where $d=\frac{1\times10^6}{1000}nm=10^3 nm$ and $n=1$

$\sin\theta=\frac{300}{10^3}=0.3$

Hence, answer is (B)

4. The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. The conventional cell has volume $a^3$. What is the volume of the primitive unit cell?
1. $\frac{a^3}{8}$
2. $\frac{a^3}{4}$
3. $\frac{a^3}{2}$
4. $a^3$
The primitive vectors of BCC structure are $$\bar a_1=a\hat x$$ $$\bar a_2=a\hat y$$ $$\bar a_3=\frac{a}{2}\left(\hat x+\hat y+\hat z\right)$$ Hence, volume of primitive cell is $\frac{a^3}{2}$

Hence, answer is (C)

5. If the absolute temperature of a blackbody is increased by a factor of 3, the energy radiated per second per unit area does which of the following
1. Decreases by a factor of 81.
2. Decreases by a factor of 9.
3. Increases by a factor of 9.
4. Increases by a factor of 81.

According to Stefan's Law the radiated power density(W/m2) of a black body is proportional to fourth power of its absolute temperature $$E=e\sigma T^4$$ The emissivity $e=1-R$ is a correction for an approximate black body radiator, where $R$ is the fraction of the light reflected. For perfectly black body $R=0$ $$\frac{E}{E'}=\frac{T^4}{T'^4}$$

Hence, answer is (D)