Polarization

Step 1: Analyze the light passing through polarizers.
When unpolarized light passes through a polarizer, the intensity is reduced by half: Next, the light passes through a quarter-wave plate with its optical axis at 45° to the polarization axis of . A quarter-wave plate changes the polarization of the light without affecting the intensity. After passing through the quarter-wave plate, the light remains polarized but is now at a 45° angle to its original polarization direction. Step 2: Analyze the second polarizer.
The second polarizer has its axis perpendicular to the initial polarization direction of the light. As the light enters the second polarizer, the intensity is reduced by a factor of . Therefore, the intensity after passing through is: Step 3: Conclusion.
Thus, the ratio is:

Step 1: Understanding the polarization of the dielectric sphere.
For a dielectric sphere with constant polarization, the bound surface charge density is given by , where is the unit normal to the surface. Since the polarization is along the z-axis, the bound surface charge density is . Outside the sphere, the electric field behaves like a dipole field, which decays as at large distances. The electric potential at on the z-axis can be derived using the appropriate formula, yielding .
Step 2: Conclusion.
Thus, the correct answers are options (A), (C), and (D).

Step 1: Understanding the effects of the plates.
The polarizer only transmits light in one direction. The plate changes the polarization from linear to elliptical depending on its orientation. The plate converts linear polarization into circular polarization when the light is incident at 45° to the optical axis. Based on the orientations, the output polarizations are determined.
Step 2: Conclusion.
Thus, the correct answer is option (A).

Step 1: Analyze the light passing through polarizers.
When unpolarized light passes through a polarizer, the intensity is reduced by half: Next, the light passes through a quarter-wave plate with its optical axis at 45° to the polarization axis of . A quarter-wave plate changes the polarization of the light without affecting the intensity. After passing through the quarter-wave plate, the light remains polarized but is now at a 45° angle to its original polarization direction.
Step 2: Analyze the second polarizer.
The second polarizer has its axis perpendicular to the initial polarization direction of the light. As the light enters the second polarizer, the intensity is reduced by a factor of . Therefore, the intensity after passing through is: Step 3: Conclusion.
Thus, the ratio is:

Step 1: Understanding the polarization of the dielectric sphere.
For a dielectric sphere with constant polarization, the field outside the sphere behaves like the field of a dipole. However, at distances large compared to the radius, the electric field falls off as , just like the field of a dipole. The surface charge density and potential outside depend on the polarization.
Step 2: Conclusion.
Thus, the correct answer is option (B).

Step 1: Understanding the effects of the plates.
The polarizer only transmits light in one direction. The plate changes the polarization from linear to elliptical depending on its orientation. The plate converts linear polarization into circular polarization when the light is incident at 45° to the optical axis. Based on the orientations, the output polarizations are determined.
Step 2: Conclusion.
Thus, the correct answer is option (A).

Step 1: Behaviour of circularly polarised light.
Circularly polarised light becomes linearly polarised when passed through a plate.

Step 2: Behaviour of unpolarised light.
Unpolarised light becomes linearly polarised only after a polariser. The plate has no effect on it before the polariser.

Step 3: Use of analyser.
After the light becomes linearly polarised, the analyser helps check intensity variation by rotation. For circularly polarised light, the analyser shows a constant intensity after the plate. For unpolarised light (after polariser), intensity varies as .

Step 4: Conclusion.
Only arrangement (D): polariser → plate → analyser can distinguish between unpolarised and circularly polarised light.

Step 1: Understand the effect of a half-wave plate.
A half-wave ( ) plate rotates the plane of polarisation by an amount equal to , where is the angle between the incident polarisation and the fast axis.

Step 2: Use the given rotation.
Given rotation = , therefore .

Step 3: Exclude quarter-wave plates.
A quarter-wave ( ) plate changes linear polarisation to elliptical or circular, not simply rotate it; hence plates cannot produce pure rotation.

Step 4: Conclusion.
Thus, the correct wave plate is a plate and .

Step 1: Retardation condition.
For conversion between opposite circular polarizations, phase difference = radians.

Step 2: Solve for thickness .

Step 3: Conclusion.
Hence, the thickness of the retarder = 33.33 µm.

Step 1: Initial intensity after first polarizer.
Unpolarized light passing through the first polarizer transmits half the intensity:

Step 2: Use Malus's law for each subsequent polarizer.
For each polarizer making angle with the previous one:

Step 3: Conclusion.

Step 1: Behavior of a quarter-wave plate.
A quarter-wave plate introduces a phase difference of between the components of light along its fast and slow axes.
Step 2: Condition for elliptical polarization.
If the incident light’s electric field makes an angle with the fast axis, its components along fast and slow axes are different. When the amplitudes differ, the resulting polarization becomes elliptical.
Step 3: Condition for circular polarization.
For circular polarization, the amplitudes must be equal ( ) and the phase difference must be . Thus, any deviation from results in elliptical polarization, but for an exact quarter-wave plate and linearly polarized input, elliptical polarization occurs precisely at .
Step 4: Final Answer.
The correct angle is