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The focal length of a mirror can be determined entirely from the shape of the mirror. In contrast, to determine the focal length of a lens, we must know both the shape of the lens and its index of refraction - as well as the index of refraction of the surrounding medium. For instance, when a thin lens is immersed in a liquid, we must modify the thin-lens equation to take into account the refractive properties of the surrounding liquid: $$\frac{1}{f}=\left(\frac{n}{n_{\mathrm{liq}}}-1\right)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right),$$ where \(n_{\text {liq }}\) is the index of refraction of the liquid and \(n\) is the index of refraction of the glass. If you place a glass lens \((n=1.5),\) which has a focal length of \(0.5 \mathrm{~m}\) in air, into a tank of water \((n=1.33),\) what will happen to its focal length? A. Nothing will happen. B. The focal length of the lens will be reduced. C. The focal length of the lens will be increased. D. There is not enough information to answer the question.

Step by step solution

01

Understanding Parameters

We are given a glass lens with a refractive index \(n = 1.5\) and a focal length \(f = 0.5\,\text{m}\) in air. The lens is placed in water, which has a refractive index \(n_{\text{liq}} = 1.33\).

02

Analyze the Thin-Lens Equation

The thin-lens equation needs to account for the surrounding liquid. The modified equation is: \[ \frac{1}{f'} = \left(\frac{n}{n_{\text{liq}}} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] where \(f'\) is the new focal length in the liquid.

03

Compare the Equation Components

In air, \(n_{\text{air}} = 1.0\), and the original focal length equation was: \[ \frac{1}{f} = \left(n - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]. In water, the modified component is \(\frac{n}{n_{\text{liq}}} - 1\), which is less than \(n - 1\).

04

Effect on Focal Length

Since \(\frac{n}{n_{\text{liq}}} - 1\) is smaller than \(n - 1\), the term \(\frac{1}{f'}\) becomes smaller than \(\frac{1}{f}\). This implies \(f' > f\), meaning the focal length increases.

05

Conclusion

Placing the lens in water increases its focal length. Thus, the focal length of the lens will be increased.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Focal Length

The focal length of a lens is a fundamental concept in optics. It is the distance from the lens where parallel rays of light converge to a single focal point. This length is a key determining factor in the optical power of the lens. A short focal length means that the lens has high optical power and bends light more sharply. Conversely, a longer focal length indicates lower optical power and a gentler curvature of light.
Understanding a lens's focal length helps in manipulating how images are formed, and it is essential in designing optical instruments like cameras, glasses, and microscopes.

Refractive Index

The refractive index, often denoted by the symbol $n$, measures how much a beam of light bends when it enters a material. When light moves from one medium to another, its speed changes, resulting in refraction. This index is calculated by the ratio of the speed of light in a vacuum to the speed of light in the medium.
Knowing the refractive index of a material is crucial because it impacts how much light will be refracted. For example, the refractive index of air is approximately 1.0, while glass typically has an index around 1.5. Water's refractive index is about 1.33. Each material's refractive index helps predict how it will alter the light's path.

Thin-Lens Equation

The thin-lens equation is a fundamental formula used to relate the focal length of a lens to the physical properties of the lens and its environment. The standard form of the thin-lens equation is given by: \[ \frac{1}{f} = \left(n - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] where \(R_1\) and \(R_2\) are the radii of curvatures of the lens's surfaces, and \(n\) is the refractive index of the lens.
This equation is essential for designing lenses to achieve a desired focal length by selecting appropriate materials and curvatures. In scenarios like lens immersion, modifications are made to the thin-lens equation to consider different surrounding mediums.

Lens Immersion

Lens immersion involves submerging a lens into a different material, altering its optical properties. When a lens is placed in a liquid, the surrounding medium's refractive index must be considered. This is because the velocity of light changes as it transitions between different materials, affecting the focal length of the lens.
Immersing a lens in a medium like water changes the thin-lens equation to account for the new environment. As seen in the problem, by substituting the liquid's refractive index into the equation, you can determine the new focal length in this medium.

Liquid Refraction

Liquid refraction occurs when light travels through a liquid, bending as a result of changes in the medium. This bending is defined by the refractive index of the liquid. In the case of lenses, this means that the lens’s effective focal length can be altered by the liquid it is submerged in.
Understanding liquid refraction is crucial for applications where lenses need to be used underwater or with liquid-based optical systems. This concept explains why a lens might focus light differently under water compared to in the air, due to a relative change in refractive indices between the glass and the liquid, affecting image formation.