How Is the Magnetic Field Calculated in the Normal Zeeman Effect?

[dinospamoni]

Homework Statement

The red line of the Balmer series in hydrogen has a
wavelength of 656.5 nm. Suppose that this line is observed to
split into three different spectral lines when placed in a magnetic field, B, due to the Normal Zeeman Effect. What is the
value of the magnetic field if the wavelength splitting between
adjacent lines is 0.065 nm?

Homework Equations

$\mu_{B}\ =\ 9.27400899(37)\ \times\ 10^{-24}\ J\ T^{-1}$

$E=E_0+\mu_B\times B\times m_l$

The Attempt at a Solution

I said that E - E0 = .065 nm

and solved for B, saying m_l = 1, but that's not right.

Any ideas?

 

[mfb]

E is an energy, nm is a length and not an energy. You can convert the wavelength values to energy values, then it will work.

 

[dinospamoni]

Ah right good catch. I tried that and got 3.3*10^8 T, but that was wrong too.

i did $E = \frac{hc}{\lambda}$

where $\lambda$ is .065 nm

than divided by $\mu_B$

 

[TSny]

Note that $E = \frac{hc}{\lambda}$ does not imply that $\Delta E = \frac{hc}{\Delta \lambda}$.

 

[Nickie]

The Zeeman effect is a phenomenon in which the spectral lines of an atom split into multiple lines when placed in a magnetic field. This is due to the interaction between the magnetic field and the magnetic moment of the atom. In this problem, we are dealing with the Normal Zeeman effect, where the spectral lines split into three lines.

To solve this problem, we can use the equation for the energy levels in a magnetic field, E = E0 + \mu_B \times B \times m_l, where \mu_B is the Bohr magneton and m_l is the magnetic quantum number. In this case, m_l can be either -1, 0, or 1, corresponding to the three spectral lines observed.

We are given that the wavelength splitting between adjacent lines is 0.065 nm. Since we know the relationship between energy and wavelength (E = hc/\lambda), we can set up the following equation:

\frac{hc}{\lambda_2} - \frac{hc}{\lambda_1} = E_2 - E_1

We can substitute the values for \lambda_2 and \lambda_1, and rearrange the equation to solve for the energy difference, E_2 - E_1. This will give us the value of \mu_B \times B \times m_l. Since we are dealing with the m_l = 1 case, we can simply plug in this value and solve for B.

The resulting value of B will be in units of Tesla (T), so we can convert it to the appropriate units (e.g. Gauss) if needed. This will give us the value of the magnetic field required to observe the splitting of the spectral line.