Diameter of a plano-convex lens is 6 cm and thickness at the centre is 3 mm. If speed of light in ma

Q: Why is the radius of the plane surface considered infinity?

A plane surface is perfectly flat, meaning it has no curvature. You can think of it as a sphere that is so unimaginably large that its surface appears flat. Therefore, its radius is mathematically treated as infinity.

Q: Why did we convert 3 mm to 0.3 cm?

Consistency in units is absolutely critical in physics. Since the diameter was given in centimeters and the final options are in centimeters, converting the thickness to centimeters prevents massive calculation errors.

Q: Is it always safe to ignore the $t^2$ term?

It is safe only when the lens is "thin," meaning its thickness $t$ is significantly smaller than its aperture radius $r$. In this case, $t^2 = 0.09$ is 100 times smaller than $r^2 = 9$, making it negligible.

Q: What if the lens was biconvex instead of plano-convex?

If it were biconvex, both surfaces would have a finite radius of curvature. You would need to calculate or be given both $R_1$ and $R_2$ and apply the proper sign convention in the Lens Maker's formula.

Q: Why is the focal length positive?

A plano-convex lens is thicker at the center than at the edges, making it a converging lens. By sign convention, converging lenses always have a positive focal length.

Question

Diameter of a plano-convex lens is $6 	ext{ cm}$ and thickness at the centre is $3 	ext{ mm}$. If speed of light in material of lens is $2 	imes 10^8 	ext{ m/s}$, the focal length of the lens is

Options:

(A) $15 	ext{ cm}$

(B) $20 	ext{ cm}$

(C) $30 	ext{ cm}$

(D) $10 	ext{ cm}$

Sigma Percentile · Accepted Answer

The journey to finding the focal length of this plano-convex lens is a beautiful blend of optics and geometry. Let's break it down step-by-step.

### The Speed of Light and Refractive Index
Before we can use any lens formulas, we need to know the optical properties of the material. The problem gives us the speed of light inside the lens, $v = 2 	imes 10^8 	ext{ m/s}$.

We know that the refractive index $n$ is the ratio of the speed of light in a vacuum $c$ to the speed of light in the medium $v$.
$$n = \frac{c}{v}$$
Substituting the standard value of $c = 3 	imes 10^8 	ext{ m/s}$:
$$n = \frac{3 	imes 10^8}{2 	imes 10^8} = 1.5$$
This tells us the lens is likely made of standard glass.

### Unlocking the Geometry of the Lens
A plano-convex lens is essentially a slice of a sphere. The curved surface has a radius of curvature $R$, and the flat surface forms a circular aperture of radius $r$. The thickness of the lens at its center is $t$.

By drawing a cross-section, we can form a right-angled triangle connecting the center of the sphere, the edge of the lens, and the center of the flat surface. Using the Pythagorean theorem:
$$R^2 = r^2 + (R - t)^2$$
Expanding this equation:
$$R^2 = r^2 + R^2 - 2Rt + t^2$$
$$2Rt = r^2 + t^2$$
Here is a crucial insight: the thickness $t = 3 	ext{ mm} = 0.3 	ext{ cm}$ is very small compared to the aperture radius $r = 3 	ext{ cm}$. Therefore, the $t^2$ term ($0.09$) is negligible compared to $r^2$ ($9$). We can safely approximate:
$$2Rt \approx r^2 \implies R \approx \frac{r^2}{2t}$$
Substituting our values:
$$R \approx \frac{3^2}{2 	imes 0.3} = \frac{9}{0.6} = 15 	ext{ cm}$$

### The Lens Maker's Formula
Now we have all the pieces to find the focal length $f$. The Lens Maker's Formula states:
$$\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} ight)$$
For a plano-convex lens, one surface is curved ($R_1 = R = 15 	ext{ cm}$) and the other is perfectly flat, meaning its radius of curvature is infinite ($R_2 = \infty$).
$$\frac{1}{f} = (1.5 - 1) \left( \frac{1}{15} - \frac{1}{\infty} ight)$$
Since $\frac{1}{\infty} = 0$:
$$\frac{1}{f} = 0.5 	imes \frac{1}{15} = \frac{1}{30}$$
Inverting both sides, we get our final answer:
**$f = 30 	ext{ cm}$**

Animated Solution for Physics - Optics: Diameter of a plano-convex lens is $6 cm$ and thickness at the centre is $3 mm$ . If speed of light in material of lens is $2 \times 1 0^{8} m/s$ , the focal length of the lens is

Select Answer:

Visualized Solution

Geometry of Plano-Convex Lens

Refractive Index

Calculating Refractive Index

Radius of Curvature Geometry

Radius of Curvature Approximation

Calculating $R$

Lens Maker's Formula

Calculating Focal Length

The Sigma Insight: Lens

The Speed of Light and Refractive Index Before we can use any lens formulas, we need to know the optical properties of the material

Unlocking the Geometry of the Lens A plano-convex lens is essentially a slice of a sphere

The Lens Maker's Formula Now we have all the pieces to find the focal length $f$

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