A paperweight is made of a solid glass hemisphere with an index of refraction of 1.50 and a radius of 4.0 cm. If placed on its flat surface above a 2.5 mm long line, what length of... A paperweight is made of a solid glass hemisphere with an index of refraction of 1.50 and a radius of 4.0 cm. If placed on its flat surface above a 2.5 mm long line, what length of the line is seen by someone looking vertically down?
Understand the Problem
The question is asking for the apparent length of a line viewed through a glass hemisphere, which involves understanding the effect of refraction on light passing through the medium. We will use Snell's law and geometry to solve for the perceived length.
Answer
The apparent length can be calculated using the formula $L' = L \cdot \frac{\sin(\theta_2)}{\sin(\theta_1)}$. The specific values depend on the given $L$ and $\theta_1$.
Answer for screen readers
The final answer for the apparent length of the line viewed through the glass hemisphere will depend on the specific values of $L$ and $\theta_1$ provided in the problem. The formula for the apparent length is: $$ L' = L \cdot \frac{\sin(\theta_2)}{\sin(\theta_1)} $$
Steps to Solve
- Identify the angles and refractive indices
According to Snell's law, the relationship between the angles and the refractive indices can be described as: $$ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) $$ Here, assume $n_1$ is the refractive index of air (approximately 1.00), and $n_2$ is the refractive index of the glass (approximately 1.5).
- Calculate the angle of refraction
Using the angles, we can substitute into Snell's law. If the angle of incidence $\theta_1$ is known, rearranging gives us: $$ \theta_2 = \sin^{-1}\left(\frac{n_1}{n_2} \sin(\theta_1)\right) $$ This will allow us to find the angle through which the light bends when moving from air to glass.
- Apply trigonometry to find the apparent length
Once $\theta_2$ is known, we will use similar triangles to relate the actual length of the line $L$ with the apparent length $L'$ as viewed through the hemisphere. The apparent length can be expressed in terms of trigonometric ratios: $$ L' = L \cdot \frac{\sin(\theta_2)}{\sin(\theta_1)} $$
- Combine all calculations to find the final answer
After determining $\theta_2$, plug the values back into the equation to find the apparent length $L'$. Ensure you substitute any actual lengths or angles you have at this stage to get a numerical result.
The final answer for the apparent length of the line viewed through the glass hemisphere will depend on the specific values of $L$ and $\theta_1$ provided in the problem. The formula for the apparent length is: $$ L' = L \cdot \frac{\sin(\theta_2)}{\sin(\theta_1)} $$
More Information
Refraction occurs due to a change in the speed of light as it crosses different mediums. This bending of light causes objects to appear at different lengths. The phenomenon is crucial in optics, forming the basis for lenses and optical devices.
Tips
- Not converting angles to radians if necessary when using trigonometric functions.
- Neglecting to account for the change in medium when applying Snell's law.
- Incorrectly tracing the angles through the hemisphere, leading to miscalculations.
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