Question

3. (i) In the presence of an external uniform magnetic field, the normal Zeeman effect is observed

and the original transition generates into three transitions of frequencies:

ν = ν0 −



eB

4πm

; ν = ν0; ν = ν0 +



eB

4πm

where m is the mass of the electron. Using a diagram, indicate what are allowed normal

Zeeman transitions between d(l = 2) state and p(l = 1) state. [6 marks]

(ii) Find the minimum magnetic field needed for the Zeeman effect to be observed in a

spectral line of 400 nm wavelength when a spectrometer whose resolution is 0.010 nm is

used.

Answer 1

the presence of an external uniform magnetic field, the normal Zeeman effect is observed

Photon energy $E=h\upsilon$

In presence of magnetic field 3 spectrum line correspondingPhoton energy

We can written as

$E_1=E_0-\mu_B B$

$E_2=E_0$

$E_3=E_0+\mu_B B$

Now frequency

$\upsilon_1=\upsilon_0-\frac{\mu_B B}{h}$

$\upsilon_2=\upsilon_0$

$\upsilon_2=\upsilon_0+\frac{\mu_B B}{h}$

$\frac{\mu_B}{h}=\frac{e}{4\pi m}$

$\upsilon_1=\upsilon_0-\frac{eB}{4\pi m} (when ∆m=-1)$

$\upsilon_2=\upsilon_0(when∆m=0)$

$\upsilon_3=\upsilon_0+\frac{eB}{4\pi m}(∆m=+1)$

We that

$\nu=\frac{c}{\lambda}$

$\nu=\frac{3\times10^8}{400\times10^{-9}}=7.5\times10^{-14}Hz$

$\nu_1=\frac{c}{\lambda_1}$

$\nu_1=\frac{3\times10^8}{400\times10^{-9}+0.010\times10^{-9}}$

$\nu_1=\frac{3\times10^8}{400\times10^{-9}}\times\frac{1}{1+\frac{0.1}{400}}$

$∆\nu=7.5\times10^{14}(1-2.5\times10^{-5})$

$∆\nu=1.875\times10^{10}Hz$

$B=\frac{4\pi m∆\nu}{e}$

$$B=\frac{4\times3.14\times9.1\times10^{-31}\times1.875\times10^{10}}{1.06\times10^{-19}}=1.34T$$