Question

The wavelength of first line of Balmer series of hydrogen atom is \( 6563\ \text{\AA} \). Determine the wavelength of second line.
OR
What do you mean by polarization of light? How are unpolarized and plane polarized lights represented? Which nature of light waves is proved by polarization?

Hint:

For the Balmer series use \( \frac{1}{\lambda} = R\left(\frac{1}{4} - \frac{1}{n^2}\right) \) with \( n = 3 \) for the first line and \( n = 4 \) for the second, then take the ratio. For the OR part, polarization confining vibrations to one plane proves light is transverse.

Answer 1

Option 1: Second Balmer line by Balmer's empirical relation

Step 1: Balmer's original empirical formula writes each line of the series directly as \[ \lambda = B\,\frac{n^2}{n^2 - 4},\quad n = 3,4,5,\dots \] where \(B\) is Balmer's constant and \(n\) labels the line (\(n=3\) first line, \(n=4\) second line).
Step 2: Fix \(B\) from the given first line. For \(n=3\): \(\lambda_1 = B\dfrac{9}{9-4} = \dfrac{9B}{5}\). Given \(\lambda_1 = 6563\) Angstrom, \[ B = \frac{5}{9}\times 6563 = 3646\ \text{Angstrom}. \] Step 3: For the second line \(n=4\): \[ \lambda_2 = B\,\frac{16}{16-4} = B\,\frac{16}{12} = \frac{4B}{3}. \] Step 4: Substitute \(B = 3646\) Angstrom: \[ \lambda_2 = \frac{4}{3}\times 3646 \approx 4861\ \text{Angstrom}. \] The same value is reached without ever using the Rydberg constant.
\[\boxed{\lambda_2 \approx 4861\ \text{Angstrom}}\]

Option 2: Polarization explained through a polaroid experiment

Step 1: Think of an unpolarized beam as a mixture of transverse waves whose electric vibrations point randomly in every direction across the ray. No single direction is preferred, so the beam has no plane of vibration.
Step 2: Place a first polaroid (polarizer) in the path. It passes only the vibration component along its axis and absorbs the rest, so the transmitted beam now vibrates in one fixed plane. This one-plane beam is plane polarized light.
Step 3: Add a second polaroid (analyser) and rotate it. The transmitted intensity rises and falls, becoming zero when its axis is perpendicular to the first. A random (unpolarized) beam would show no such variation, proving the first polaroid produced polarized light.
Step 4 (Symbols): On a ray diagram, unpolarized light carries both dots (\(\odot\)) and arrows (\(\updownarrow\)) together; plane polarized light carries only arrows (\(\updownarrow\)) or only dots (\(\odot\)).
Step 5 (Conclusion): Because polarization is possible only for waves that vibrate across the direction of travel, it establishes that light is a transverse wave and not a longitudinal one.
\[\boxed{\text{Light is a transverse wave, confirmed by polarization.}}\]