Height of Image Formed by Refraction at Curved Surface

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

$\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R}$

1. Identify the given parameters using the Cartesian sign convention:

• Refractive index of the first medium (air): $\mu_1 = 1$
• Refractive index of the second medium (lens material): $\mu_2 = 1.54$
• Object position: $u = -40\text{ cm}$
• Radius of curvature: $R = -20\text{ cm}$
• Height of the object: $h_o = 2\text{ cm}$

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2. Calculate the position of the image ($v$):

Substitute the given values into the refraction formula:

$\frac{1.54}{v} - \frac{1}{-40} = \frac{1.54 - 1}{-20}$

$\frac{1.54}{v} + \frac{1}{40} = -\frac{0.54}{20}$

$\frac{1.54}{v} = -\frac{1.08}{40} - \frac{1}{40}$

$\frac{1.54}{v} = -\frac{2.08}{40}$

$v = -\frac{1.54 \times 40}{2.08} \text{ cm}$

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3. Calculate the transverse magnification ($m$):

The lateral or transverse magnification for refraction at a spherical surface is given by:

$m = \frac{h_i}{h_o} = \frac{\mu_1 v}{\mu_2 u}$

Substituting $v = -\frac{1.54 \times 40}{2.08}$ into the magnification expression:

$m = \frac{1 \times \left(-\frac{1.54 \times 40}{2.08}\right)}{1.54 \times (-40)} = \frac{1}{2.08}$

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4. Calculate the height of the image ($h_i$):

$h_i = |m| \times h_o = \frac{1}{2.08} \times 2\text{ cm} = \frac{2}{2.08}\text{ cm} \approx 0.9615\text{ cm} \approx 1\text{ cm}$

Thus, the height of the image is approximately $1\text{ cm}$.

Correct Answer: A