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: Wavelength of the second Balmer line

Step 1 (Formula): For the Balmer series of hydrogen, the wavelength is given by
\[ \frac{1}{\lambda} = R\left(\frac{1}{2^2} - \frac{1}{n^2}\right),\quad n = 3,4,5,\dots \] where \( R \) is the Rydberg constant. The first line corresponds to \( n = 3 \) and the second line to \( n = 4 \).

Step 2 (First line, n = 3):
\[ \frac{1}{\lambda_1} = R\left(\frac{1}{4} - \frac{1}{9}\right) = R\left(\frac{9-4}{36}\right) = \frac{5R}{36} \]
Step 3 (Second line, n = 4):
\[ \frac{1}{\lambda_2} = R\left(\frac{1}{4} - \frac{1}{16}\right) = R\left(\frac{4-1}{16}\right) = \frac{3R}{16} \]
Step 4 (Take the ratio): Dividing the two relations removes \( R \):
\[ \frac{\lambda_2}{\lambda_1} = \frac{5R/36}{3R/16} = \frac{5}{36}\times\frac{16}{3} = \frac{80}{108} = \frac{20}{27} \]
Step 5 (Substitute the data): With \( \lambda_1 = 6563\ \text{\AA} \),
\[ \lambda_2 = \frac{20}{27}\times 6563 = \frac{131260}{27} \approx 4861\ \text{\AA} \]
This lies in the blue-green region, which agrees with the known second Balmer (H\(_\beta\)) line.
\[\boxed{\lambda_2 \approx 4861\ \text{\AA}}\]

Option 2: Polarization of light

Step 1 (Meaning): Ordinary light coming from a source has its electric-field vibrations in all directions perpendicular to the direction of propagation. Polarization is the process of restricting these vibrations to a single plane. Light in which the vibrations are confined to only one plane is called plane (linearly) polarized light.
Step 2 (How it happens): When ordinary (unpolarized) light is passed through a polaroid or is reflected/scattered suitably, only the component of vibration parallel to the transmission axis passes through. The emerging beam vibrates in one plane and is therefore plane polarized.
Step 3 (Representation): Unpolarized light is represented on a ray by showing both dots (\( \odot \), vibrations perpendicular to the paper) and double-headed arrows (\( \updownarrow \), vibrations in the plane of the paper) together, indicating vibrations in every direction. Plane polarized light is represented by showing only arrows (\( \updownarrow \)) when the plane of vibration is in the plane of the paper, or only dots (\( \odot \)) when it is perpendicular to the paper.
Step 4 (Nature proved): Since only transverse waves (in which vibrations are perpendicular to propagation) can be polarized, and longitudinal waves cannot, the phenomenon of polarization proves that light is a transverse wave.
\[\boxed{\text{Polarization proves the transverse nature of light waves.}}\]