A coin of diameter 4cm is placed at a distance of 45cm in front of a convex lens of focal length 30cm  . Then the diameter of the image of the coin will be:

Correct option is B

Correct Answer: (B) 8 cm

Explanation: To calculate the diameter of the image, we will first use the lens formula and then the magnification formula.

Given:

• Object diameter = 4 cm
• Object distance (u) = -45 cm (negative as the object is on the same side as the incoming light)
• Focal length (f) = 30 cm

Step 1: Use the lens formula to find the image distance (v)

The lens formula is:

​$$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$​​

Substitute the given values:

​$$\frac{1}{30} = \frac{1}{v} - \frac{1}{-45}$$​​

Simplifying:

​$$\frac{1}{30} = \frac{1}{v} + \frac{1}{45}$$​​

Finding a common denominator:

​$$\frac{1}{v} = \frac{1}{30} - \frac{1}{45}\\ \ \\\frac{1}{v} = \frac{3}{90} - \frac{2}{90}\\ \ \\\frac{1}{v} = \frac{1}{90}$$​​

Therefore:

v=90 cm

Step 2: Use the magnification formula to find the image size

The magnification (M) is given by:

​$$M = \frac{\text{Image size}}{\text{Object size}} = \frac{v}{u}$$​​

Substitute the values:

​$$M = \frac{90}{-45} = -2$$​​

The negative sign indicates that the image is inverted. Now, the magnification is also the ratio of the image diameter to the object diameter:

​$$M = \frac{\text{Image diameter}}{\text{Object diameter}}$$​​

Substitute the values:

​$$-2 = \frac{\text{Image diameter}}{4}$$​​

Solving for the image diameter:

Image diameter=4×2=8 cm

Conclusion:

The diameter of the image is 8 cm.

Information Booster:

• A convex lens forms a real, inverted, and diminished/enlarged image depending on object placement.
  
• If the object is beyond 2F (twice the focal length), the image is inverted, real, and diminished.
  
• If the object is between F and 2F, the image is real, inverted, and enlarged.
  
• The image is of the same size when the object is at 2F.
  
• Convex lenses are used in magnifying glasses, cameras, and human eyes.

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