In a medium of refractive index 1.6 and having a convex surface has a point object in it at a distance of 12 cm from the pole. The radius of curvature is 6 cm. Locate the image as seen from air

Options

A. A real image at 30 cm
B. A virtual image at 30 cm
C. A real image at 4.28 cm
D. A virtual image at 4.28 cm

Answer

To solve the problem of locating the image formed by a convex surface in a medium with a refractive index of 1.6, we will use the formula for refraction at a spherical surface. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Refractive index of the medium (μ1) = 1.6
- Refractive index of air (μ2) = 1 (since we are locating the image as seen from air)
- Object distance (u) = -12 cm (the negative sign indicates that the object is on the same side as the incoming light)
- Radius of curvature (R) = +6 cm (positive for a convex surface)

### Step 2: Use the refraction formula
The formula for refraction at a spherical surface is given by:

\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]

### Step 3: Substitute the known values into the formula
Substituting the values we have:

\[ \frac{1}{v} - \frac{1.6}{-12} = \frac{1 - 1.6}{6} \]

### Step 4: Simplify the equation
Rearranging the equation gives:

\[ \frac{1}{v} + \frac{1.6}{12} = \frac{-0.6}{6} \]

Calculating \(\frac{1.6}{12}\):

\[ \frac{1.6}{12} = \frac{0.1333}{1} \]

Calculating \(\frac{-0.6}{6}\):

\[ \frac{-0.6}{6} = -0.1 \]

Now substituting these values back into the equation:

\[ \frac{1}{v} + 0.1333 = -0.1 \]

### Step 5: Solve for \(\frac{1}{v}\)
Now, isolate \(\frac{1}{v}\):

\[ \frac{1}{v} = -0.1 - 0.1333 \]
\[ \frac{1}{v} = -0.2333 \]

### Step 6: Find the value of \(v\)
Taking the reciprocal gives:

\[ v = \frac{1}{-0.2333} \approx -4.29 \text{ cm} \]

### Step 7: Interpret the result
Since \(v\) is negative, this indicates that the image is virtual and located on the same side as the object.

### Final Answer
The image is located approximately 4.29 cm to the left of the surface, indicating it is virtual.

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