Question

Object is placed at $40 \text{ cm}$ from spherical surface whose radius of curvature is $20 \text{ cm}$. Find height of image formed.

• A. $2 \text{ cm}$
• B. $4 \text{ cm}$
• C. $0.96 \text{ cm}$
• D. $1.96 \text{ cm}$

Answer 1

The Correct Option is C

To find the height of the image formed by the spherical surface, we need to use the formula for refraction at a spherical surface:

\(\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}\)

Where:

• \(\mu_1 = 1\) (refractive index of medium 1)
• \(\mu_2 = 1.54\) (refractive index of medium 2)
• \(u = -40 \text{ cm}\) (object distance, negative in Cartesian coordinates)
• \(R = 20 \text{ cm}\) (radius of curvature, positive)
• \(v\) is the image distance (what we need to find)

Substituting the values into the equation:

\(\frac{1.54}{v} - \frac{1}{-40} = \frac{1.54 - 1}{20}\)

\(\frac{1.54}{v} + \frac{1}{40} = \frac{0.54}{20}\)

Solving for \(\frac{1.54}{v}\):

\(\frac{1.54}{v} = \frac{0.54}{20} - \frac{1}{40}\)

Finding common denominator:

\(\frac{0.54 \times 2}{40} - \frac{1}{40} = \frac{1.08 - 1}{40} = \frac{0.08}{40}\)

Therefore, \(v = \frac{1.54 \times 40}{0.08} = 770 \text{ cm}\)

Now, using magnification formula:

\(m = \frac{h_i}{h_o} = \frac{\mu_1 \cdot v}{\mu_2 \cdot u}\)

• Given \(h_o = 2 \text{ cm}\)

Substituting values:

\(m = \frac{h_i}{2} = \frac{1 \cdot 770}{1.54 \cdot -40}\)

\(h_i = 2 \times \frac{1 \cdot 770}{1.54 \cdot -40} = -0.96 \text{ cm}\)

The negative sign indicates the image is inverted. Thus, the height of the image is \(0.96 \text{ cm}\).