An image is formed at a distance of 100 cm from the glass surface with refractive index 1.5, when a point object is placed in the air at a distance of 100 cm from the glass surface. The radius of curvature is of the surface is

Options

A. 20 cm
B. 40 cm
C. 30 cm
D. 50 cm

Answer

To solve the problem, we will use the formula for refraction at a spherical surface. Here are the steps to find the radius of curvature (R) of the glass surface:

### Step 1: Identify the given values
- The refractive index of glass (μ2) = 1.5
- The distance of the image from the glass surface (v) = 100 cm (positive since it's on the same side as the light travels)
- The distance of the object from the glass surface (u) = -100 cm (negative since the object is on the opposite side of the light travel)

### Step 2: Write down the refraction formula
The formula for refraction at a spherical surface is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
Where:
- μ1 = 1 (the refractive index of air)
- μ2 = 1.5 (the refractive index of glass)
- v = image distance
- u = object distance
- R = radius of curvature

### Step 3: Substitute the values into the formula
Substituting the known values into the equation:
\[ \frac{1.5}{100} - \frac{1}{-100} = \frac{1.5 - 1}{R} \]

### Step 4: Simplify the equation
Calculating the left-hand side:
\[ \frac{1.5}{100} + \frac{1}{100} = \frac{1.5 + 1}{100} = \frac{2.5}{100} \]
So, we have:
\[ \frac{2.5}{100} = \frac{0.5}{R} \]

### Step 5: Cross-multiply to solve for R
Cross-multiplying gives:
\[ 2.5R = 0.5 \times 100 \]
\[ 2.5R = 50 \]

### Step 6: Solve for R
Now, divide both sides by 2.5:
\[ R = \frac{50}{2.5} = 20 \text{ cm} \]

### Final Answer
The radius of curvature (R) of the glass surface is **20 cm**.

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