A convex lens forms a real image of an object for its two different positions on a screen.If the height of image in both the cases is 9cm and 4cm respectively then, height of the object is _____cm...Pls provide the whole method of solution.

To find the height of the object when a convex lens forms real images of different heights, we can use the lens formula and the magnification formula. Let's break this down step by step.

Understanding the Problem

We have a convex lens that produces two real images of an object at different positions. The heights of these images are given as 9 cm and 4 cm. We need to determine the height of the object itself.

Key Concepts

• Magnification (m): This is defined as the ratio of the height of the image (h') to the height of the object (h). It can be expressed as:
• m = h' / h
• Real Images: For real images formed by a convex lens, the magnification is negative, indicating that the image is inverted.

Setting Up the Equations

Let’s denote the height of the object as h. For the two positions of the object, we can write the magnification equations as follows:

• For the first image (height 9 cm):
• m₁ = -h'₁ / h = -9 / h
• For the second image (height 4 cm):
• m₂ = -h'₂ / h = -4 / h

Relating the Two Positions

Since the object is the same in both cases, we can relate the two magnifications. The magnification is also related to the distances from the lens to the object and the image. However, for simplicity, we can focus on the height ratios directly.

Setting Up the Ratio

From the magnification equations, we can express the heights in terms of the object height:

• For the first image: h = -9 / m₁
• For the second image: h = -4 / m₂

Since both expressions equal h, we can set them equal to each other:

Solving for the Object Height

Now, we can equate the two expressions:

-9 / m₁ = -4 / m₂

Cross-multiplying gives us:

9 * m₂ = 4 * m₁

Now, we can express m₁ and m₂ in terms of h:

m₁ = -9 / h and m₂ = -4 / h

Substituting these into the equation:

9 * (-4 / h) = 4 * (-9 / h)

Now, simplifying:

-36 / h = -36 / h

This equation holds true, confirming our magnification ratios are consistent. However, we need to find the actual height of the object.

Finding the Height

To find the height of the object, we can use either magnification equation. Let’s use the first one:

m₁ = -9 / h

We know that the magnification is also related to the distances, but since we have the heights, we can rearrange this to find h:

h = -9 / m₁

Since we have two magnifications, we can average them or use any specific value. However, since we have a consistent ratio, we can directly calculate:

Using the average of the heights for simplicity:

h = (9 + 4) / 2 = 6.5 cm

Final Result

Thus, the height of the object is approximately 6.5 cm. This method illustrates how we can use the properties of lenses and magnification to solve for unknown dimensions in optics.