Polariser Intensity Problem Setup
We are given three polarisers, $P_1$, $P_3$, and $P_2$. Initially, $P_1$ and $P_2$ are oriented such that the transmitted light intensity between them is zero. This implies their transmission axes are perpendicular.
Let the angle of the transmission axis of $P_1$ be $\alpha_1 = 0$. Then, the angle of the transmission axis of $P_2$ is $\alpha_2 = \frac{\pi}{2}$ (or $90^\circ$).
Calculating Maximum Transmitted Intensity
A third polariser, $P_3$, is placed between $P_1$ and $P_2$. Let its transmission axis be at an angle $\theta$ relative to $P_1$. Thus, its angle is $\alpha_3 = \theta$.
The intensity $I$ after passing through all three polarisers, starting from unpolarized light of initial intensity $I_{unpol}$, is given by:
$I = I_{unpol} \cos^2(\alpha_3 - \alpha_1) \cos^2(\alpha_2 - \alpha_3)$
Substituting the angles:
$I = I_{unpol} \cos^2(\theta - 0) \cos^2(\frac{\pi}{2} - \theta)$
Using the identity $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$, the expression becomes:
$I = I_{unpol} \cos^2(\theta) \sin^2(\theta)$
Finding the Angle for Maximum Intensity
To maximize the transmitted intensity $I$, we need to maximize the term $\cos^2(\theta) \sin^2(\theta)$.
We can rewrite this term using the double angle identity $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$:
$\cos^2(\theta) \sin^2(\theta) = (\cos(\theta)\sin(\theta))^2 = \left(\frac{\sin(2\theta)}{2}\right)^2 = \frac{1}{4} \sin^2(2\theta)$
The intensity is:
$I = I_{unpol} \frac{1}{4} \sin^2(2\theta)$
This intensity is maximum when $\sin^2(2\theta)$ is maximum, which is 1.
$\sin^2(2\theta) = 1$
This occurs when $2\theta = \frac{\pi}{2}$. Solving for $\theta$ gives:
$\theta = \frac{\pi}{4}$
This angle $\theta = \frac{\pi}{4}$ is the angle of $P_3$ relative to $P_1$. The question asks for the angle between polarisers $P_2$ and $P_3$.
Determining the Angle Between $P_2$ and $P_3$
The angle between $P_2$ and $P_3$ is $|\alpha_2 - \alpha_3|$.
Angle = $|\frac{\pi}{2} - \theta| = |\frac{\pi}{2} - \frac{\pi}{4}| = \frac{\pi}{4}$.
Therefore, the angle between the polarisers $P_2$ and $P_3$ is $\frac{\pi}{4}$.
Similar Questions
Consider following statements for refraction of light through prism, when angle of deviation is minimum.
A. The refracted ray inside prism becomes parallel to the base.
B. Larger angle prisms provide smaller angle of minimum deviation.
C. Angle of incidence and angle of emergence becomes equal.
D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
E. Angle of refraction becomes double of prism angle. Choose the correct answer from the options given below:
The radii of curvature for a thin convex lens are $10 \ cm$ and $15 \ cm$ respectively. The focal length of the lens is $12 \ cm$. The refractive index of the lens material is
The work function of a metal is $3 \ eV$. The color of the visible light that is required to cause emission of photoelectrons is
In the figure shown below, a resistance of $150.4 \Omega$ is connected in series to an ammeter A of resistance $240 \Omega$. A shunt resistance of $10 \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :
A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure. Distance of the image of an object 'O', is :
(C is the center of curvature of the spherical surface and R is the radius of curvature)- If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is $30^\circ$ in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is _________ $\mu$m.
- A finite size object is placed normal to the principal axis at a distance of 30 cm from a convex mirror of focal length 30 cm. A plane mirror is now placed in such a way that the image produced by both the mirrors coincide with each other. The distance between the two mirrors is:
- In a Young's double slit experiment, two slits are located 1.5 mm apart. The distance of screen from slits is 2 m and the wavelength of the source is 400 nm. If the 20 maxima of the double slit pattern are contained within the central maximum of the single slit diffraction pattern, then the width of each slit is $x\times10^{-3}$ cm, where x-value is ________.
- Two thin convex lenses of focal lengths $30 \text{ cm}$ and $10 \text{ cm}$ are placed coaxially, $10 \text{ cm}$ apart. Thepower of this combination is:
Important Questions from Optics
Consider following statements for refraction of light through prism, when angle of deviation is minimum.
A. The refracted ray inside prism becomes parallel to the base.
B. Larger angle prisms provide smaller angle of minimum deviation.
C. Angle of incidence and angle of emergence becomes equal.
D. There are always two sets of angle of incidence for which deviation will be same except at minimum deviation setting.
E. Angle of refraction becomes double of prism angle. Choose the correct answer from the options given below:
The radii of curvature for a thin convex lens are $10 \ cm$ and $15 \ cm$ respectively. The focal length of the lens is $12 \ cm$. The refractive index of the lens material is
The work function of a metal is $3 \ eV$. The color of the visible light that is required to cause emission of photoelectrons is
In the figure shown below, a resistance of $150.4 \Omega$ is connected in series to an ammeter A of resistance $240 \Omega$. A shunt resistance of $10 \Omega$ is connected in parallel with the ammeter. The reading of the ammeter is __________ mA.
A slanted object AB is placed on one side of convex lens as shown in the diagram. The image is formed on the opposite side. Angle made by the image with principal axis is :
