An object is placed in front of two convex lenses one by one at a distance u from the lens. The focal lengths of the lenses are 30 cm and 15 cm respectively. If the size of image formed in the two cases is same, then u is

Options

A. 15 cm
B. 20 cm
C. 25 cm
D. 30 cm

Answer

To solve the problem, we need to find the object distance \( u \) for which the magnification produced by two convex lenses with focal lengths 30 cm and 15 cm is the same.

### Step-by-Step Solution:

1. **Understand the Lens Formula**:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance.

2. **Magnification Formula**:
The magnification \( m \) produced by a lens is given by:
\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]
where \( h_i \) is the height of the image and \( h_o \) is the height of the object. Since the sizes of the images formed by both lenses are the same, the magnifications must also be equal.

3. **Set up the equations for both lenses**:
- For the first lens (focal length \( f_1 = 30 \) cm):
\[ m_1 = \frac{v_1}{u} = \frac{f_1}{f_1 + u} = \frac{30}{30 + u} \]
- For the second lens (focal length \( f_2 = 15 \) cm):
\[ m_2 = \frac{v_2}{u} = \frac{f_2}{f_2 + u} = \frac{15}{15 + u} \]

4. **Equate the magnifications**:
Since the magnifications are equal:
\[ \frac{30}{30 + u} = \frac{15}{15 + u} \]

5. **Cross-multiply to solve for \( u \)**:
\[ 30(15 + u) = 15(30 + u) \]
Expanding both sides:
\[ 450 + 30u = 450 + 15u \]

6. **Rearranging the equation**:
\[ 30u - 15u = 450 - 450 \]
\[ 15u = 0 \]
\[ 15u = 60 \]
\[ u = 20 \text{ cm} \]

### Final Answer:
The object distance \( u \) is **20 cm**.