Question

(a) Figure 33-27 shows light reaching a polarizing sheet whose polarizing direction is parallel to a y axis. We shall rotate the sheet 40° clockwise about the light’s indicated line of travel. During this rotation, does the fraction of the initial light intensity passed by the sheet increase, decrease, or remain the same if the light is (a) initially unpolarized, (b) initially polarized parallel to the x axis, and (c) initially polarized parallel to the y axis?

Short answer

(a) The fraction of initial light intensity passed by the sheet if the light is initially unpolarized will remain the same.

(b) The fraction of initial light intensity passed by the sheet if the light is initially polarized parallel to the x-axis will increase by a factor ${\cos }^2\theta$.

(c) The fraction of initial light intensity passed by the sheet if the light is initially polarized parallel to the y-axis will decrease by a factor ${\cos }^2\theta$.

Step-by-step solution

The given data

The polarization direction of the sheet is initially parallel with the y axis.

The sheet is rotated $40°$clockwise.

Understanding the concept of the intensity of polarization

Light can be polarized by passing it through a polarizing filter or other polarizing material. The intensity of an unpolarized light after passing through a polarized filter is given by the cosine-square rule. Here, we need to use the concept of polarization using polarizing sheets along with the equations for the intensity of light passing through the polarizing sheet.

Formulae:

If the incident light is un-polarized, then the intensity of the merging light using the one-half rule is given by,

$I=0.5I_0$ (i)

If the incident light is already polarized, then the intensity of the emerging light is cosine –squared of the intensity of incident light,

$I=I_0{\cos }^2\theta$ (ii)

Here $\theta$is the angle between the polarization of the incident light and the polarization axis of the sheet.

a) Calculation of the initial light intensity if the light is initially unpolarized

When the incident light is initially un-polarized, the intensity of light emerging from the sheet is given by equation (i), that is $I=0.5I_0$.

As, the intensity does not depend on the angle; thus, the fraction of intensity will remain the same.

b) Calculation of the initial light intensity if the light is initially polarized parallel to the x-axis

As the incident light is initially polarized parallel to the x-axis and the polarizing direction of the sheet is parallel to the y-axis. No light can pass through the sheet. But when we rotate the sheet, some light can pass as a component and is given by equation (ii) as $I=I_0{\cos }^2\theta$:

So, the fraction of light is increased by ${\cos }^2\theta$.

c) Calculation of the initial light intensity if the light is initially polarized parallel to the y-axis

As the incident light is initially polarized parallel to the y-axis, and the polarizing direction of the sheet is also parallel to the y-axis. All light can pass through the sheet.

But when we rotate the sheet, some light can be blocked, and the intensity of light emerging out is given using equation (ii) by, $I=I_0{\cos }^2\theta$

So, the fraction of the light decreases by ${\cos }^2\theta$.