Q32P - Fundamentals Of Physics

Question

Figure  is a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is placed in a magnetic field of $\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{ }\mathit{T}{\mathit{E}}_{\mathbf{2}}$ , the graph changes to that of Figure b because of the energy associated with ${\overrightarrow{\mathbf{\mu }}}_{\mathbf{o}\mathbf{r}\mathbf{b}}\mathbf{.}\overrightarrow{\mathbf{B}}$. Level ${\mathit{E}}_{\mathbf{1}}$ is unchanged, but level splits into a (closely spaced) triplet of levels. What are the allowed values of  ${\mathit{m}}_{\mathbf{1}}$ associated with (a) Energy level ${\mathit{E}}_{\mathbf{1}}$ and (b) Energy level  ${\mathit{E}}_{\mathbf{2}}$? (c) In joules, what amount of energy is represented by the spacing between the triplet levels?

Step-by-Step Solution

Answer

a) Allowed values of $m_l$  associated with energy level $E_1$ is zero.

b) Allowed values of $m_l$  associated with energy level $E_2$ as upper line has $m_l=+1$ and lower will be $m_l=-1$

c) Amount of energy in joules represented by the spacing between the triplet levels is $4.64\times {10}^{-24 }\mathrm{J}$.

Step 1: Listing the given quantities

Magnetic field,  $B_{ext}=0.500 \mathrm{T}$

Step 2: Understanding the concepts of magnetic dipole moment

To calculate the  $m_l$values, we can use the formula of energy associated with the orientation of an orbital magnetic dipole moment in an external field. Substituting the values of $∆m_l$, Bohr magnetron, and the magnetic field, we can calculate the spacing between the energy levels.

Formula:

$U=-m_l{\mu }_B{B}_{ext}$

Step 3: (a) Calculations of the allowed values of m l associated with energy level E 1

The energy  U associated with the orientation of the orbital magnetic dipole moment in an external magnetic field  ${\overrightarrow{B}}_{ext}$ is

$\begin{array}{rcl}U& =& -{\overrightarrow{\mu }}_{orb}.{\overrightarrow{B}}_{ext}\\ & =& -{\mu }_{orb,z}{B}_{ext}\end{array}$

Since

${\mu }_{orb,z}=-m_l\frac{eh}{4\pi m}$

Also,

$\frac{eh}{4\pi m}={\mu }_B$

Thus,

${\mu }_{orb,z}=-m_l{\mu }_B$

Therefore,

$\begin{array}{rcl}U& =& -{\overrightarrow{\mu }}_{orb}.{\overrightarrow{B}}_{ext}\\ & =& -m_l{\mu }_B{B}_{ext}\end{array}$

From the figure, we can observe that there is no splitting of energy level $E_1$  in the external magnetic field.

Hence its energy does not change by the application of external magnetic field ${\overrightarrow{B}}_{ext}$

Hence $U\propto m_l=0$

$\therefore m_l=0$

Step 4: (b) Calculations of allowed values of m l associated with energy level E 2

The energy level $E_2$  splits into three energy levels, with the middle one unshifted from the original value of energy. $E_2$Hence its $m_l$ value is zero.

In other two lines, one line is shifted above the unshifted line and another one is shifted below.

Hence the upper line has $m_l=+1$ and lower will be $m_l=-1$

 

Step 5: (c) Calculations ofamount of energy in joules represented by the spacing between the triplet levels

The spacing between the energy levels is given by

$\begin{array}{rcl}\Delta U& =& {\mu }_B{B}_{ext}\\ & =& \left|\Delta \left(-m_l{\mu }_B{B}_{ext}\right)\right|\end{array}$

For any pair of adjacent levels in triplet,

$\left|\Delta m_l\right|=1$

Therefore,

where, ${\mu }_B$  is Bohr magnetron, and its value is ${\mu }_B=9.27\times {10}^{-24} \mathrm{J}/\mathrm{T}$

$\begin{array}{rcl}\Delta U& =& 9.27\times {10}^{-24}\frac{\mathrm{J}}{\mathrm{T}}\times 0.500 \mathrm{T}\\ & =& 4.64\times {10}^{-24}\mathrm{J}\end{array}$

Amount of energy in joules represented by the spacing between the triplet levels is $4.64\times {10}^{-24} \mathrm{J}$.