Question

Suppose that a hydrogen atom in its ground state moves 80 cm through and perpendicular to a vertical magnetic field that has a magnetic field gradient$\frac{\mathbf{dB}}{\mathbf{dz}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{2}}\mathbf{T}$ . (a) What is the magnitude of force exerted by the field gradient on the atom due to the magnetic moment of the atom’s electron, which we take to be Bohr magnetron? (b) What is the vertical displacement of the atom in the 80cm of travel if its speed is $\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}$?

Short answer

1. The magnitude of the force exerted by the field gradient on the atom due to the magnetic moment is $1.5\times {10}^{-21}\mathrm{N}$.
2. The vertical displacement of the atom is $2\times {10}^{-5}\mathrm{m}$.

Step-by-step solution

The given data:

1. The gradient value of the magnetic field, $\frac{\mathrm{dB}}{\mathrm{dz}}=1.6\times {10}^2\mathrm{T}/\mathrm{m}$
2. The horizontal displacement of the atom in its round state, $\mathrm{r}=80\mathrm{cm}=0.8\mathrm{m}$
3. The speed of the atom, $\mathrm{v}=1.2\times {10}^5\mathrm{m}/\mathrm{s}$

Understanding the concept of the Stern-Gerlach experiment:

Definitions of gravitational force. The force of attraction between all masses in the cosmos, particularly the attraction of the earth's mass to things near its surface.

Use the concept of gravitational force, to find gravitational potential energy, and integrate the equation of gravitational force over infinity to reference position. Then find the work required to increase the separation of the particles forthegiven positions.

Formulas:

The magnitude of the force exerted on an atom in the magnetic field,

$\mathrm{F}={\mathrm{\mu }}_{\mathrm{B}}\frac{\mathrm{dB}}{\mathrm{dz}}$ ….. (1)

Here, the Bohr magneton is ${\mathrm{\mu }}_{\mathrm{B}}=9.27\times {10}^{-24}\mathrm{J}/\mathrm{T}$.

The second equation of kinematic motion starting from rest,

$∆\mathrm{x}=\frac{1}{2}{\mathrm{at}}^2$ ….. (2)

Here, $∆\mathrm{x}$is the change in the position,a is he acceleration, and t is time.

The time taken for travel is defined by,

$\mathrm{t}=\frac{\mathrm{r}}{\mathrm{v}}$ ….. (3)

The force due to Newton’s second law,

F = ma ….. (4)

Here, m is the mass.

(a) Calculation of the magnitude of the exerted force:

Using the given data in equation (1), the force exerted on the hydrogen atom due to the presence of a magnetic field as follows:

$$\begin{gathered} \mathrm{F}=\left(9.27\times {10}^{-24}\right)\left(1.6\times {10}^2\right) \\ =1.5\times {10}^{-21}\mathrm{N} \end{gathered}$$

Hence, the value of the exerted force is $1.5\times {10}^{-21}\mathrm{N}$.

(b) Calculation of the vertical displacement of the atom:

Substituting the value of time and acceleration from, equations (3) and (4) respectively in equation (2), the vertical displacement of the atom is as follow.

(For mass of hydrogen, $\mathrm{m}=1.6\times {10}^{-27}\mathrm{kg}$)

$$\begin{gathered} ∆\mathrm{x}=\frac{1}{2}\left(\frac{\mathrm{F}}{\mathrm{m}}\right){\left(\frac{\mathrm{r}}{\mathrm{v}}\right)}^2 \\ =\frac{1}{2}\left[\frac{\left(1.5\times {10}^{-21}\mathrm{N}\right){\left(0.8\mathrm{m}\right)}^2}{\left(1.6\times {10}^{-27}\mathrm{kg}\right){\left(1.2\times {10}^5\mathrm{m}/\mathrm{s}\right)}^2}\right] \\ =2\times {10}^{-5}\mathrm{m} \end{gathered}$$

Hence, the value of the vertical displacement is $2\times {10}^{-5}\mathrm{m}$.