A plano convex lens of refractive index 1.5 and radius of curvature 30cm. Is silvered at the curved surface. Now this lens has been used to form the image of an object. At what distance from this lens an object be placed in order to have a real image of size of the object.

Options

A. `tan^(-1)((t_(1))/t_(2))`
B. `sin^(-1)((t_(1))/t_(2))`
C. `sin^(-1)((10t_(1))/t_(2))`
D. `tan^(-1)((10t_(1))/t_(2))`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Calculate the Focal Length of the Plano-Convex Lens

The formula for the focal length \( f_1 \) of a plano-convex lens is given by:

\[ \frac{1}{f_1} = \left( \mu - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( \mu \) is the refractive index of the lens (1.5).
- \( R_1 \) is the radius of curvature of the first surface (infinity for plano surface).
- \( R_2 \) is the radius of curvature of the second surface (30 cm, which is taken as -30 cm because it is a convex surface).

Substituting the values:

\[ \frac{1}{f_1} = (1.5 - 1) \left( \frac{1}{\infty} - \frac{1}{-30} \right) \]

This simplifies to:

\[ \frac{1}{f_1} = 0.5 \left( 0 + \frac{1}{30} \right) = \frac{0.5}{30} = \frac{1}{60} \]

Thus,

\[ f_1 = 60 \text{ cm} \]

### Step 2: Determine the Focal Length of the Silvered Surface

Since the lens is silvered at the curved surface, it behaves like a concave mirror. The focal length \( f_m \) of a concave mirror is given by:

\[ f_m = -\frac{R}{2} \]

Where \( R \) is the radius of curvature (30 cm).

Thus,

\[ f_m = -\frac{30}{2} = -15 \text{ cm} \]

### Step 3: Calculate the Equivalent Focal Length

The equivalent focal length \( f_{eq} \) of the system (lens + mirror) can be calculated using the formula:

\[ \frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_m} \]

Substituting the values we found:

\[ \frac{1}{f_{eq}} = \frac{1}{60} + \frac{1}{-15} \]

Finding a common denominator (which is 60):

\[ \frac{1}{f_{eq}} = \frac{1}{60} - \frac{4}{60} = -\frac{3}{60} = -\frac{1}{20} \]

Thus,

\[ f_{eq} = -20 \text{ cm} \]

### Step 4: Determine the Object Distance for a Real Image of Same Size

For a real image of the same size as the object, the object distance \( u \) must be equal to twice the focal length:

\[ u = 2f_{eq} \]

Substituting the value of \( f_{eq} \):

\[ u = 2 \times (-20) = -40 \text{ cm} \]

Since the object distance is conventionally taken as positive in the direction of the incoming light, we take the absolute value:

\[ u = 40 \text{ cm} \]

### Final Answer

The object should be placed at a distance of **40 cm** from the lens to form a real image of the same size as the object.

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