A thin convex lens made from crown glass (mu = 3/2) has focal length f. When it is measured in two d

Q: Why does the focal length change when the lens is put in a liquid?

The bending of light depends on the *difference* in optical density between the lens and its surroundings. A liquid reduces this difference compared to air, altering the bending power and thus the focal length.

Q: What does a negative focal length physically mean for a convex lens?

It means the lens is now diverging light instead of converging it. The rays spread out, and they appear to originate from a virtual focus behind the lens.

Q: Can a concave lens act as a converging lens?

Absolutely! If you place a concave lens in a medium that is optically denser than the lens material itself, it will converge light.

Q: Why did we substitute $(1/R_1 - 1/R_2)$ with $1/x$?

Since the physical shape of the lens doesn't change when moved between liquids, this geometric term remains constant. Substituting it with a single variable simplifies the algebra significantly.

Q: Is the focal length in liquid always greater than in air?

Yes, as long as the liquid is rarer than the lens material. The relative refractive index decreases, which decreases the bending power, leading to a longer focal length.

Question

A thin convex lens made from crown glass $\left(\mu = \frac{3}{2}\right)$ has focal length $f$. When it is measured in two different liquids having refractive indices $\frac{4}{3}$ and $\frac{5}{3}$. It has the focal lengths $f_1$ and $f_2$, respectively. The correct relation between the focal lengths is

Options:

(A) $f_1 = f_2 < f$

(B) $f_1 > f$ and $f_2$ becomes negative

(C) $f_2 > f$ and $f_1$ becomes negative

(D) $f_1$ and $f_2$ both become negative

Sigma Percentile · Accepted Answer

### The Magic of the Lens Maker's Formula

Imagine you are holding a beautiful convex lens made of crown glass. In the air, it behaves exactly as you'd expect, converging parallel rays of light to a crisp focal point.

But what happens when we plunge this lens into different liquids? Does it still converge light the same way?

To answer this, we need our master key: the **Lens Maker's Formula**.

$$ \frac{1}{f} = \left( \frac{\mu_g}{\mu_m} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $$

This elegant equation tells us that the focal length $f$ isn't just about the shape of the lens. It depends heavily on the **relative refractive index** between the glass ($\mu_g$) and the surrounding medium ($\mu_m$).

Let's first establish our baseline in the air. The refractive index of our glass is $3/2$, and air is simply $1$.

$$ \frac{1}{f} = \left( \frac{3/2}{1} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $$

$$ \frac{1}{f} = \frac{1}{2} \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $$

To keep our algebra clean, let's bundle that geometric curvature term $(1/R_1 - 1/R_2)$ into a single variable, $1/x$. So, our baseline equation becomes $1/f = 1/(2x)$, which means $f = 2x$.

### Diving into the First Liquid

Now, let's submerge our lens into the first liquid, which has a refractive index of $4/3$.

The glass is still optically denser than the liquid ($3/2 > 4/3$), so the lens should still converge light. But let's see exactly how much.

$$ \frac{1}{f_1} = \left( \frac{3/2}{4/3} - 1 \right) \frac{1}{x} $$

$$ \frac{1}{f_1} = \left( \frac{9}{8} - 1 \right) \frac{1}{x} $$

$$ \frac{1}{f_1} = \frac{1}{8x} $$

Notice how the multiplier dropped from $1/2$ to $1/8$. Since $1/(2x)$ was our original $1/f$, we can rewrite this as:

$$ \frac{1}{f_1} = \frac{1}{4} \left( \frac{1}{2x} \right) = \frac{1}{4f} $$

This gives us **$f_1 = 4f$**. The focal length has increased dramatically! The lens is still converging, but it's much weaker now, bending the rays far less than it did in the air. Therefore, we know for sure that **$f_1 > f$**.

### The Plot Twist: A Denser Medium

Next, we move the lens into the second liquid. This liquid has a refractive index of $5/3$.

Here is where the physics gets thrilling. The liquid ($5/3 \approx 1.67$) is actually **optically denser** than the crown glass ($3/2 = 1.5$)!

Let's plug this into our formula and see what the math reveals.

$$ \frac{1}{f_2} = \left( \frac{3/2}{5/3} - 1 \right) \frac{1}{x} $$

$$ \frac{1}{f_2} = \left( \frac{9}{10} - 1 \right) \frac{1}{x} $$

$$ \frac{1}{f_2} = -\frac{1}{10x} $$

We have a negative sign! This is a profound physical result.

### The Final Verdict

A negative focal length means that our convex lens has completely changed its nature. It is no longer converging light; it is now **diverging** light, behaving exactly like a concave lens.

So, we have established two critical facts. First, in the lighter liquid, the focal length increased ($f_1 > f$). Second, in the denser liquid, the focal length became negative.

This perfectly aligns with our theoretical understanding and leads us directly to the correct conclusion.

**Final Answer: $f_1 > f$ and $f_2$ becomes negative.**

Select Answer:

Visualized Solution

$L e n s in A i r$

$L e n s M ak e r^{'} s F or m u l a$

$F oc a l L e n g t h in A i r$

$L e n s in L i q u i d 1$

$F oc a l L e n g t h in L i q u i d 1$

$C o m p a r in g f_{1} an d f$

$L e n s in L i q u i d 2$

$F oc a l L e n g t h in L i q u i d 2$

$N a t u r e o f L e n s in L i q u i d 2$

$C o n c l u s i o n$

The Sigma Insight: Lens

The Magic of the Lens Maker's Formula

Diving into the First Liquid

The Plot Twist

The Final Verdict

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Lens in Air

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Lens Maker′s Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Focal Length in Air

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Lens in Liquid 1

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Focal Length in Liquid 1

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Comparing f1​ and f

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Lens in Liquid 2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Focal Length in Liquid 2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Nature of Lens in Liquid 2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion