Q.Light from a point source in air falls on a spherical glass surface ($n = 1.5$ and radius of curvature $= 20\ \text{cm}$). The distance of the light source from the glass surface is $100\ \text{cm}$. At what position the image is formed?

Concept: Refraction at a single spherical surface.

Reasoning:

1. Use the standard formula for refraction at a spherical surface:

where $n_1 = 1$ (air), $n_2 = 1.5$ (glass), $u = -100\ \text{cm}$ (real object, negative sign convention), and $R = +20\ \text{cm}$ (convex surface towards the object).

2. Substitute the values:

$$ \frac{1.5}{v} + \frac{1}{100} = \frac{0.5}{20} = \frac{1}{40} $$ …

The problem is a classic application of refraction at a single spherical surface. Light travels from air (refractive index $n_1 = 1$) into glass ($n_2 = 1.5$). The surface is convex toward the incident light (the source is in air, the glass is on the other side). The radius of curvature $R = +20\ \text{cm}$ by the sign convention: the centre of curvature lies on the side where light goes (into the glass), so $R$ is positive.

The key formula that governs this is the single spherical surface equation, which relates object distance $u$, image distance $v$, the two refractive indices, and the radius of curvature.

Here, $n_1 = 1$ (air), $n_2 = 1.5$ (glass), $R = +20\ \text{cm}$, and $u = -100\ \text{cm}$ (object is real, on the incident side, so $u$ is negative by the Cartesian sign convention).

Let’s work through it step by step.

1. Set up the equation with signs. The object is real and placed in front of the surface, so $u = -100\ \text{cm}$. The surface is convex toward the object, so $R = +20\ \text{cm}$. Substitute into the formula:

2. Simplify the right-hand side.

3. Simplify the left-hand side. The term $-\frac{1}{-100}$ becomes $+\frac{1}{100}$:

4. Isolate $\frac{1.5}{v}$.

Find a common denominator (200):

So:

5. Solve for $v$.

$v = \frac{300}{3} = 100\ \text{cm}$ …

Method: Refraction Formula for a Single Spherical Surface

We use the Gaussian formula for refraction at a spherical surface:

Step-by-step solution

Step 1: Identify the given values

• Refractive index of air, $n_1 = 1$
• Refractive index of glass, $n_2 = 1.5$
• Radius of curvature, $R = +20\ \text{cm}$ (convex surface toward the object in air)
• Object distance from surface, $u = -100\ \text{cm}$ (negative by sign convention — object is real and on the incident side)

Step 2: Apply the formula

Step 3: Simplify

Common Mistakes & How to Avoid Them: Refraction at Spherical Surface

Mistake 1: Wrong Sign Convention for $R$

The error: Students often take $R = +20\ \text{cm}$ without checking the surface orientation.

In this problem, light travels from air to glass through a convex surface (since the source is in air and the centre of curvature lies on the other side).

Correct approach:

• For a convex surface (centre of curvature on the transmitted side), $R$ is positive.
• Here, $R = +20\ \text{cm}$ is correct — but only if you verify the direction.

How to avoid:

Always draw a ray diagram. Mark the centre of curvature $C$. If $C$ lies on the side where light is going after refraction, $R > 0$. If it lies on the incident side, $R < 0$.

Mistake 2: Confusing $n_1$ and $n_2$ in the Formula

The error: Swapping refractive indices in the formula:

Students sometimes write $\frac{n_1}{v} - \frac{n_2}{u}$ or misplace $n_1$ and $n_2$.

Correct assignment:

• $n_1$ = refractive index of the incident medium = 1 (air)
• $n_2$ = refractive index of the transmitted medium = 1.5 (glass)

How to avoid:

Write the formula as:

Always label $n_1$ and $n_2$ before plugging numbers.

Mistake 3: Forgetting the Sign of $u$

The error: Using $u = +100\ \text{cm}$ instead of $u = -100\ \text{cm}$.

Correct approach:

By Cartesian sign convention, object distance $u$ is negative when the object is on the incident side (real object).

So $u = -100\ \text{cm}$.

How to avoid:

Remember: real object → $u$ is negative. Only virtual objects (rare in basic problems) give positive $u$.

Mistake 4: Arithmetic Errors in Solving for $v$

The error: Rushing through the algebra and making sign mistakes.

Correct calculation:

How to avoid:

• Write each step clearly.
• Keep fractions — avoid decimals until the final step. …