Chapter 6: Q62P (page 1405)

Electrons in the lower of two spin states in a magnetic field can absorb a photon of the right frequency and move to the higher state. (a) Find the magnetic field magnitude Brequired for this transition in a hydrogen atom with n= 1 and l= 0 to be induced by microwaves with wavelength $λ$ . (b) Calculate the value of Bfor a wavelength of 4.20 cm.

Expert verified

The magnetic field magnitude for the given transition is $B = \frac{2 πmc}{λe}$ .
The value of Bfor a wavelength of 4.20 cm is 0.255 T

Step by step solution

The potential energy is given as,

$U = ‐ μ_{z} B$ ...(i)

Where, B is the effective magnetic field and is given as,

$μ_{z} = - (2.00232) (\frac{e}{2 m}) S_{z}$

Here, $S_{z} = h$

Thus, the energy difference is given as,

$∆ E = (2.00232) (\frac{e}{2 m}) B ∆ S_{z} \approx \frac{e h}{m} B = \frac{hc}{λ}$

Solving for the magnetic field B is,

$\Rightarrow B = \frac{2 πmc}{λe}$

From the expression for the magnetic field obtained in part (a),

$B = \frac{2 π (9.11 \times 10^{- 31} kg) (3.00 \times 10^{8} m / s)}{(0.0420 m) (1.60 \times 10^{- 19} C)} = 0.255 T$

Thus the magnitude of magnetic field is 0.255 T

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