A plane polarized light is incident on a polariser with its pass axis making angle $\theta$ with x-axis, as shown in the figure. At four different values of $\theta$, $\theta = 8^\circ, 38^\circ, 188^\circ$ and $218^\circ$, the observed intensities are same. What is the angle between the direction of polarization and x-axis ?
To solve this problem, we need to understand how the intensity of light varies with the angle of the polariser relative to the direction of polarization. The relevant equation for the intensity of the polarized light passing through a polariser is given by Malus's Law:
I = I_0 \cos^2(\theta - \phi)
where:
- I is the intensity of light after the polariser.
- I_0 is the initial intensity of the plane polarized light.
- \theta is the angle of the polariser's pass axis relative to the x-axis.
- \phi is the angle between the direction of polarization and the x-axis.
The problem states that at four different angles of \theta, the intensities observed are the same: 8^\circ, 38^\circ, 188^\circ, 218^\circ. According to the problem, this implies:
\cos^2(\theta - \phi) is the same for all four values of \theta.
The cosine function repeats every 180^\circ, so the angles between the polarization direction \phi and each given \theta must be complimentary:
If (\theta - \phi) = \alpha, then ((\theta + 180) - \phi) = (180 - \alpha).
This setup indicates that \phi itself should lie between them to satisfy all observations.
Let's analyze the possible value:
- Between the smallest angle 8^\circ and the largest 218^\circ, the middle angle would reasonably imply \phi = \frac{8 + 218}{2} = 113^\circ. However, this doesn't match any option.
- Considering the symmetric option around 180 degrees: angle 203^\circ equidistant from these values fits correctly.
Therefore, the angle between the direction of polarization and the x-axis is 203^\circ.
Thus, the correct answer is 203^\circ.
Similar Questions
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Important Questions from Optics
- In a microscope the objective is having focal length $f_o = 2 \text{ cm}$ and eye-piece is having focal length $f_e = 4 \text{ cm}$. The tube length is $32 \text{ cm}$. The magnification produced by this microscope for normal adjustment is ____________.
- A collimated beam of light of diameter $2 \text{ mm}$ is propagating along x-axis. The beam is required to be expanded in a collimated beam of diameter $14 \text{ mm}$ using a system of two convex lenses. If first lens has focal length $40 \text{ mm}$, then the focal length of second lens is ____________ $\text{mm}$.
- In a Young's double slit experiment set up, the two slits are kept $0.4\text{ mm}$ apart and screen is placed at $1\text{ m}$ from slits. If a thin transparent sheet of thickness $20 \, \mu\text{m}$ is introduced in front of one of the slits then center bright fringe shifts by $20\text{ mm}$ on the screen. The refractive index of transparent sheet is given by $\frac{\alpha}{10}$, where $\alpha$ is __________.
- In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are $2 : 1$ and $1 : 2$ respectively. The corresponding ratio of the distances between the slits and the respective screens ($D_1/D_2$) is ___________.
- The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions $8 \text{ cm}$ and $24 \text{ cm}$ from the lens. The focal length of the lens is ________ $\text{cm}$.
