Final answer:
To find, the angle of incidence in the glass, we can use Snell's law. The angle of incidence in the glass can be calculated using the formula: sin(angle of incidence in glass) / sin(angle of refraction in ethanol) = index of refraction of ethanol / index of refraction of flint glass.
So, the correct answer is:
a) \( \theta_1 \approx 21.49^\circ \)
b) \( \theta_1 \approx 21.49^\circ \)
c) \( \theta_1 \approx 21.49^\circ \)
d) \( \theta_1 \approx 21.49^\circ \)
Step-by-step explanation:
To solve this problem, we can use Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media. Snell's Law is given by:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
where:
- \( n_1 \) is the index of refraction of the first medium (flint glass),
- \( \theta_1 \) is the angle of incidence in the first medium,
- \( n_2 \) is the index of refraction of the second medium (ethanol),
- \( \theta_2 \) is the angle of refraction in the second medium.
In this case:
- \( n_1 = 1.61 \) (index of refraction for flint glass),
- \( \theta_2 = 26.1^\circ \) (angle of refraction in ethanol),
- \( n_2 = 1.36 \) (index of refraction for ethanol).
We want to find \( \theta_1 \), the angle of incidence in the flint glass.
First, let's rearrange Snell's Law to solve for \( \theta_1 \):
\[ \theta_1 = \sin^{-1}\left(\frac{n_2}{n_1} \sin(\theta_2)\right) \]
Now, plug in the values:
\[ \theta_1 = \sin^{-1}\left(\frac{1.36}{1.61} \sin(26.1^\circ)\right) \]
Now, calculate \( \theta_1 \). Make sure your calculator is set to degrees mode.
Certainly, let's continue with the calculation:
\[ \theta_1 = \sin^{-1}\left(\frac{1.36}{1.61} \sin(26.1^\circ)\right) \]
Now, substitute the values and compute the expression:
\[ \theta_1 = \sin^{-1}\left(\frac{1.36}{1.61} \times \sin(26.1^\circ)\right) \]
\[ \theta_1 \approx \sin^{-1}\left(\frac{1.36}{1.61} \times 0.43837\right) \]
\[ \theta_1 \approx \sin^{-1}(0.37052) \]
Use the arcsine function on your calculator to find the angle:
\[ \theta_1 \approx 21.49^\circ \]
Now that we have the angle of incidence (\( \theta_1 \)), you can select the appropriate answer format. In this case, it is given in degrees. So, the correct answer is:
a) \( \theta_1 \approx 21.49^\circ \)