The magnification of an object plac ed in front of a convex lens of focal length 20 cm is +2. to obtain a magnification of -2. the object will have to be moved a distance equal to

Options

A. 10 cm
B. 20 cm
C. 30 cm
D. 40 cm

Answer

To solve the problem, we need to determine the distance the object must be moved to change the magnification from +2 to -2 using a convex lens with a focal length of 20 cm.

### Step 1: Understand the magnification formula
The magnification (m) produced by a lens is given by the formula:
\[ m = \frac{f}{f + u} \]
where \( f \) is the focal length and \( u \) is the object distance.

### Step 2: Calculate the initial object distance (u1) for magnification +2
Given:
- \( m_1 = +2 \)
- \( f = +20 \, \text{cm} \)

Using the magnification formula:
\[ 2 = \frac{20}{20 + u_1} \]

Cross-multiplying gives:
\[ 2(20 + u_1) = 20 \]
\[ 40 + 2u_1 = 20 \]
\[ 2u_1 = 20 - 40 \]
\[ 2u_1 = -20 \]
\[ u_1 = -10 \, \text{cm} \]

### Step 3: Calculate the new object distance (u2) for magnification -2
Now, we want to find the object distance when the magnification is -2:
- \( m_2 = -2 \)

Using the magnification formula again:
\[ -2 = \frac{20}{20 + u_2} \]

Cross-multiplying gives:
\[ -2(20 + u_2) = 20 \]
\[ -40 - 2u_2 = 20 \]
\[ -2u_2 = 20 + 40 \]
\[ -2u_2 = 60 \]
\[ u_2 = -30 \, \text{cm} \]

### Step 4: Calculate the distance moved by the object
The distance the object must be moved is given by:
\[ \text{Distance moved} = u_2 - u_1 \]
Substituting the values we found:
\[ \text{Distance moved} = -30 - (-10) \]
\[ \text{Distance moved} = -30 + 10 \]
\[ \text{Distance moved} = -20 \, \text{cm} \]

### Conclusion
The object must be moved 20 cm to the left (indicated by the negative sign).

### Final Answer
The object will have to be moved a distance equal to **20 cm**.

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