In displacement method distance of object from convex lens of focal length 20 cm in one position 60 cm. Then

Options

A. in the other position distance of object from convex lens will be 30 cm
B. distance between object and screen is 90 cm
C. Both (a) and (b) are correct
D. Both (a) and (b) are wrong

Answer

To solve the problem using the displacement method for a convex lens, we will follow these steps:

### Step 1: Identify the given data
- Focal length of the convex lens (f) = 20 cm
- Object distance in the first position (u₁) = -60 cm (the negative sign indicates that the object is on the opposite side of the light direction)

### Step 2: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) = focal length
- \( v \) = image distance
- \( u \) = object distance

Substituting the known values into the lens formula:
\[ \frac{1}{20} = \frac{1}{v} - \frac{1}{-60} \]
This simplifies to:
\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{60} \]

### Step 3: Solve for v
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{20} - \frac{1}{60} \]
To solve this, find a common denominator (which is 60):
\[ \frac{1}{20} = \frac{3}{60} \]
So:
\[ \frac{1}{v} = \frac{3}{60} - \frac{1}{60} = \frac{2}{60} = \frac{1}{30} \]
Thus, we find:
\[ v = 30 \text{ cm} \]

### Step 4: Calculate the distance between the object and the screen
The distance between the object and the screen (D) is given by:
\[ D = |u| + |v| = 60 + 30 = 90 \text{ cm} \]

### Step 5: Determine the other position of the object
Let the object distance in the second position be \( u_2 \). The image distance in this case will be:
\[ v_2 = D - u_2 = 90 - u_2 \]

Using the lens formula again:
\[ \frac{1}{f} = \frac{1}{v_2} - \frac{1}{u_2} \]
Substituting \( v_2 \):
\[ \frac{1}{20} = \frac{1}{90 - u_2} - \frac{1}{u_2} \]

### Step 6: Solve for u₂
Rearranging gives:
\[ \frac{1}{20} = \frac{u_2 - (90 - u_2)}{u_2(90 - u_2)} \]
This simplifies to:
\[ \frac{1}{20} = \frac{2u_2 - 90}{u_2(90 - u_2)} \]
Cross-multiplying leads to:
\[ u_2(90 - u_2) = 20(2u_2 - 90) \]
Expanding both sides:
\[ 90u_2 - u_2^2 = 40u_2 - 1800 \]
Rearranging gives:
\[ u_2^2 - 50u_2 + 1800 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:
\[ u_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -50, c = 1800 \):
\[ u_2 = \frac{50 \pm \sqrt{(-50)^2 - 4 \cdot 1 \cdot 1800}}{2 \cdot 1} \]
Calculating the discriminant:
\[ u_2 = \frac{50 \pm \sqrt{2500 - 7200}}{2} \]
This gives:
\[ u_2 = \frac{50 \pm \sqrt{2500}}{2} = \frac{50 \pm 30}{2} \]
The two solutions are:
\[ u_2 = \frac{80}{2} = 40 \quad \text{and} \quad u_2 = \frac{20}{2} = 10 \]
However, since we are looking for the second position where the object distance is 30 cm, we find:
\[ u_2 = 30 \text{ cm} \]

### Final Answer
Both statements in the options are correct:
- The distance of the object from the convex lens in the other position is 30 cm.
- The distance between the object and the screen is 90 cm.