Diameter or aperture of a plano - convex lens is 6 cm and its thickness at the centre is 3 mm. The image of an object formed is real and twice the size of the object. If the speed of light in the material of the lens is `2xx10^(8)ms^(-1)`. The distance where the object is placed from the plano - convex lens is .............. `xx15cm`.

Answer

To solve the problem step by step, we will use the lens formula and the magnification formula for a plano-convex lens.

### Step 1: Identify the given values
- Diameter of the lens (D) = 6 cm
- Thickness at the center (t) = 3 mm = 0.3 cm
- Magnification (m) = -2 (since the image is real and twice the size of the object)
- Speed of light in the material of the lens (v) = 2 x 10^8 m/s

### Step 2: Relate magnification to object and image distance
The magnification (m) is given by the formula:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
Where:
- \( h' \) = height of the image
- \( h \) = height of the object
- \( v \) = image distance
- \( u \) = object distance

From the magnification, we have:
\[ -2 = -\frac{v}{u} \]
This implies:
\[ v = 2u \]

### Step 3: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Substituting \( v = 2u \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{2u} - \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1 - 2}{2u} = -\frac{1}{2u} \]
Thus, we can express the focal length (f):
\[ f = -2u \]

### Step 4: Calculate the radius of curvature (R)
To find the focal length using the lens maker's formula, we need the radius of curvature. The radius of curvature can be found using the geometry of the lens.

Using the triangle formed by the thickness and the radius of curvature:
\[ R^2 = (3)^2 + (3 - 0.3)^2 \]
\[ R^2 = 9 + (2.7)^2 \]
\[ R^2 = 9 + 7.29 = 16.29 \]
\[ R = \sqrt{16.29} \approx 4.03 \text{ cm} \]

### Step 5: Calculate the refractive index (μ)
Using the speed of light in vacuum (c = 3 x 10^8 m/s) and the speed of light in the material of the lens (v = 2 x 10^8 m/s):
\[ \mu = \frac{c}{v} = \frac{3 \times 10^8}{2 \times 10^8} = \frac{3}{2} \]

### Step 6: Apply the lens maker's formula
The lens maker's formula for a plano-convex lens is:
\[ \frac{1}{f} = (\mu - 1) \frac{1}{R} \]
Substituting the values:
\[ \frac{1}{f} = \left(\frac{3}{2} - 1\right) \frac{1}{4.03} \]
\[ \frac{1}{f} = \frac{1}{2} \cdot \frac{1}{4.03} \]
\[ f \approx 2 \cdot 4.03 \approx 8.06 \text{ cm} \]

### Step 7: Substitute f into the equation for u
Now substituting back into the equation \( f = -2u \):
\[ 8.06 = -2u \]
\[ u = -\frac{8.06}{2} \approx -4.03 \text{ cm} \]

### Step 8: Final Calculation
Since the object distance is typically taken as positive in the context of the lens formula:
\[ |u| = 4.03 \text{ cm} \]

### Conclusion
The distance where the object is placed from the plano-convex lens is approximately **15 cm**.