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Chapter 1: Problem 88

Suppose you put on two pairs of polarizing sunglasses with their axes at an angle of \(15.0^{\circ} .\) How much longer will it take the light to deposit a given amount of energy in your eye compared with a single pair of sunglasses? Assume the lenses are clear except for their polarizing characteristics.

Short Answer

Expert verified
The light will take \(\frac{1}{\cos^2(15^\circ)}\) times longer to deposit the same amount of energy in the eye when wearing two pairs of sunglasses with their axes at an angle of \(15.0^{\circ}\) compared to a single pair of sunglasses.

Step by step solution

01

Understand Malus' Law

Malus' Law states that the intensity (I) of transmitted light through a polarizer is given by the equation: \[ I = I_0 \cdot \cos^2(\theta) \] where \(I_0\) is the intensity of the incident polarized light, \(\theta\) is the angle between the polarizing axis of the sunglasses and the direction of polarization of the incident light.
02

Calculate the intensity of light transmitted through the first pair of sunglasses

As the exercise states, the axes of the sunglasses are at an angle of 15 degrees. When the light passes through the first pair of sunglasses, the intensity will be reduced according to Malus' Law. Let's assume the incident light has intensity \(I_0\), so after passing through the first pair of sunglasses, the transmitted light intensity will become: \[ I_{1} = I_0 \cdot \cos^2(0^\circ) \] Since \(\cos(0) = 1\), we have: \[ I_{1} = I_0\]
03

Calculate the intensity of light transmitted through the second pair of sunglasses

Now, let's focus on the light transmitted through the second pair of sunglasses, which are at an angle of \(15.0^{\circ}\) to the first pair. The intensity of the light transmitted through the second pair of sunglasses will be: \[ I_{2} = I_{1} \cdot \cos^2(15^\circ) \] We already found that \(I_{1} = I_0\), so we can rewrite as: \[ I_{2} = I_0 \cdot \cos^2(15^\circ) \]
04

Calculate the factor of how much longer it will take for light to deposit a given amount of energy

Let \(t\) be the time it takes for light to deposit a given amount of energy in the eye with a single pair of sunglasses, and let \(t'\) be the time it takes with both pairs of sunglasses. We are given that the intensity of light with one pair of sunglasses is \(I_0\) and with two pairs of sunglasses is \(I_2\). The energy can be found as the product of intensity and time, hence: \[ I_0 \cdot t = I_2 \cdot t' \] Since we want to know how much longer it will take to deposit the same amount of energy, we can find the ratio between \(t'\) and \(t\): \[ \frac{t'}{t} = \frac{I_0}{I_2} \] Now, replacing \(I_2\) with the expression we found earlier: \[ \frac{t'}{t} = \frac{I_0}{I_0 \cdot \cos^2(15^\circ)} \]
05

Find the final answer

We can simplify and find the ratio between the two times: \[ \frac{t'}{t} = \frac{1}{\cos^2(15^\circ)} \] Thus, the light will take \(\frac{1}{\cos^2(15^\circ)}\) times longer to deposit the same amount of energy in the eye when wearing two pairs of sunglasses with their axes at an angle of \(15.0^{\circ}\) compared to a single pair of sunglasses.

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