A 3.0 cm object is placed 8.0 cm in front of a mirror.The virtual image is 4.0 cm further from the mirror when the mirror is concave than when it is planar.
Determine the focal length of the concave mirror.

Options

A. 6.0 cm
B. 24 cm
C. 12 cm
D. 48 cm

Answer

To solve the problem step by step, we will follow the information provided in the question and apply the mirror formula.

### Step 1: Identify the Object Distance
We are given that the object distance (u) is 8.0 cm in front of the mirror. According to the sign convention for mirrors, distances measured in the direction of the incident light (towards the mirror) are negative. Therefore:
\[ u = -8.0 \, \text{cm} \]

### Step 2: Determine the Image Distance for the Plane Mirror
For a plane mirror, the image distance (v) is equal to the object distance but on the opposite side of the mirror. Thus:
\[ v_{\text{plane}} = +8.0 \, \text{cm} \]

### Step 3: Determine the Image Distance for the Concave Mirror
According to the problem, the virtual image formed by the concave mirror is 4.0 cm further from the mirror than the image formed by the plane mirror. Therefore, the image distance for the concave mirror is:
\[ v_{\text{concave}} = v_{\text{plane}} + 4.0 \, \text{cm} = 8.0 \, \text{cm} + 4.0 \, \text{cm} = 12.0 \, \text{cm} \]
Since this image is virtual and behind the mirror, we take it as positive:
\[ v = +12.0 \, \text{cm} \]

### Step 4: Apply the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values we have:
\[ \frac{1}{f} = \frac{1}{12.0} + \frac{1}{-8.0} \]

### Step 5: Calculate the Right Side of the Equation
Calculating the right side:
\[ \frac{1}{f} = \frac{1}{12} - \frac{1}{8} \]
To combine these fractions, we need a common denominator, which is 24:
\[ \frac{1}{f} = \frac{2}{24} - \frac{3}{24} = \frac{-1}{24} \]

### Step 6: Solve for the Focal Length
Taking the reciprocal gives us the focal length:
\[ f = -24.0 \, \text{cm} \]

### Conclusion
The focal length of the concave mirror is:
\[ \boxed{-24.0 \, \text{cm}} \]

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