Problem set 72

1. In the circuit given below, the thermister has a resistance 3 $k\Omega$ at $25^o C$. Its resistance decreases by 150 $\Omega$ per $^oC$ upon heating. The output voltage of the circuit at $30^oC$ is
   
1. $-3.75$ V
2. $-2.25$ V
3. $2.25$ V
4. $3.75$ V

Resistance of thermister at $30^oC$ is $R_f=2.25$ \begin{align*} V_{out}&=-\frac{R_f}{R_i}V_{in}\\ &=-\frac{2.25}{1}(-1)\\ &=2.25\:V \end{align*}

Hence, answer is (C)

2. Of the following term symbols of the $np^2$ atomic configurations, $^1S_0$, $^3P_0$, $^3P_1$, $^3P_2$ and $^1D_2$, which is the ground state?
1. $^3P_0$
2. $^1S_0$
3. $^3P_2$
4. $^3P_1$

According to Hund’s rules

1. State with highest multiplicity has lowest energy
2. State with same multiplicity, the state with highest $L$ will have lowest energy
3. State with same multiplicity and $L$ value. The state with lowest $J$ has lowest energy (only if subshell is less than half filled)

From the given states $^1S_0$, $^3P_0$, $^3P_1$, $^3P_2$ and $^1D_2$, $^3P_0$ will have the lowest energy

Hence, answer is (A)

3. A diatomic molecule has vibrational states with energies $E_\nu = \hbar\omega \left(\nu+\frac{1}{2}\right)$ and rotational states with energies $E_j = Bj(j + 1)$, where $\nu$ and $j$ are non-negative integers. Consider the transitions in which both the initial and final states are restricted to $\nu\leq 1$ and $j\leq 2$ and subject to the selection rules $\Delta\nu = \pm 1$ and $\Delta j = \pm 1$. Then the largest allowed energy of transition is
1. $\hbar\omega-3B$
2. $\hbar\omega-B$
3. $\hbar\omega+4B$
4. $2\hbar\omega+B$

$$E=\hbar\omega \left(\nu+\frac{1}{2}\right)+Bj(j + 1)$$ where $j$ is lower quantum number For $\Delta\nu = \pm 1$ and $\Delta j = \pm 1$ $$\Delta E=E_i-E_f=\hbar\omega+2B(j + 1)$$ For the largest allowed energy of transition $j=1$ $$\Delta E=E_i-E_f=\hbar\omega+4B$$

Hence, answer is (C)

4. A He-Ne laser operates by using two energy levels of Ne separated by 2.26 eV. Under steady state conditions of optical pumping, the equivalent temperatures of the system at which the ratio of the number of atoms in the upper state to that in the lower state will be 1/20, is approximately (the Boltzman constant $k_B=8.6\times10^{-5}\:eV/K$
1. $10^{10}\:K$
2. $10^{8}\:K$
3. $10^{6}\:K$
4. $10^{4}\:K$

$$\frac{N_2}{N_1}=e^{-E/k_BT}$$ \begin{align*} T&=-\frac{E}{k_B}\frac{1}{\ln{\frac{N_2}{N_1}}}\\ &=-\frac{2.26}{8.6\times10^{-5}}\frac{1}{\ln{\frac{1}{20}}}\\ &\approx 10^{4} \end{align*}

Hence, answer is (D)

5. The critical magnetic fields of a superconductor at temperatures 4K and 8K are 11mA/m and 5.5 mA/m respectively. The transition temperature is approximately
1. 8.4 K
2. 10.6 K
3. 12.9 K
4. 15.0K

$$H_{c_T}=H_{c_0}\left(1-\left(\frac{T}{T_c}\right)^2\right)$$ $$H_{c_{T_1}}=H_{c_0}\left(1-\left(\frac{T_1}{T_c}\right)^2\right)$$ $$H_{c_{T_2}}=H_{c_0}\left(1-\left(\frac{T_2}{T_c}\right)^2\right)$$ $$\frac{H_{c_{T_1}}}{H_{c_{T_2}}}=\frac{T_c^2-T_1^2}{T_c^2-T_2^2}$$ $$\frac{T_c^2-16}{T_c^2-64}=\frac{11}{5.5}$$ $$T_c=10.6 \:K$$

Hence, answer is (B)