A beam of light converges at a point P. Now a concave lens of focal length -16 cm is placed in the path of the convergent beam 12 cm from P The point at which the beam converges now is

Options

A. 6.86cm right side of the lens
B. 6.86cm left side of the lens
C. 48cm right side of the lens
D. 48cm left side of the lens

Explanation

To solve the problem step by step, we will use the lens formula and the information provided about the concave lens and the position of the light rays.

### Step 1: Understand the situation
A concave lens with a focal length (f) of -16 cm is placed 12 cm away from the point P where the light rays were initially converging. We need to find the new point of convergence after the light passes through the lens.

### Step 2: Identify the sign convention
In this scenario, we will use the following sign convention:
- Distances measured in the direction of the incoming light (towards the lens) are considered positive.
- The focal length of a concave lens is negative.

### Step 3: Assign values
- Focal length of the lens, \( f = -16 \) cm
- Distance from the lens to point P, \( u = +12 \) cm (since it is measured in the direction of the incoming light)

### Step 4: Use the lens formula
The lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
Where:
- \( v \) = image distance (the distance from the lens to the new point of convergence)
- \( u \) = object distance (distance from the lens to the point P)

Substituting the known values into the lens formula:
\[ \frac{1}{v} - \frac{1}{12} = \frac{1}{-16} \]

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{12} - \frac{1}{16} \]

### Step 6: Finding a common denominator
To perform the subtraction, we need a common denominator. The least common multiple of 12 and 16 is 48. Therefore:
\[ \frac{1}{12} = \frac{4}{48} \quad \text{and} \quad \frac{1}{16} = \frac{3}{48} \]
Now substituting these values:
\[ \frac{1}{v} = \frac{4}{48} - \frac{3}{48} = \frac{1}{48} \]

### Step 7: Calculate \( v \)
Now, taking the reciprocal to find \( v \):
\[ v = 48 \text{ cm} \]

### Step 8: Determine the direction
Since \( v \) is positive, it indicates that the new point of convergence is on the opposite side of the lens from the incoming light, which is to the right of the lens.

### Final Answer
The new point at which the beam converges is 48 cm to the right of the lens.

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