To determine where to place the screen for a sharp image from a concave mirror, we first need to find the focal length of the mirror and then use the mirror formula to find the image distance. Let's break this down step by step.
Finding the Focal Length
The focal length (f) of a concave mirror is related to its radius of curvature (R) by the formula:
f = R / 2
Given that the radius of curvature is 36 cm, we can calculate the focal length:
f = 36 cm / 2 = 18 cm
Using the Mirror Formula
The mirror formula relates the object distance (u), image distance (v), and focal length (f) as follows:
1/f = 1/v + 1/u
Here, the object distance (u) is given as -27 cm (the negative sign indicates that the object is in front of the mirror). Now, we can substitute the values into the mirror formula:
1/18 = 1/v + 1/(-27)
Solving for Image Distance
Rearranging the equation gives us:
1/v = 1/18 + 1/27
To add these fractions, we need a common denominator, which is 54:
Now, adding these fractions:
1/v = 3/54 + 2/54 = 5/54
Taking the reciprocal gives:
v = 54/5 = 10.8 cm
Positioning the Screen
To obtain a sharp image, the screen should be placed at a distance of 10.8 cm from the mirror. This distance is positive, indicating that the image is formed on the same side as the object, which is typical for concave mirrors when the object is placed beyond the focal point.
Nature and Size of the Image
The image formed by a concave mirror in this scenario will be:
- Real: Since the object is beyond the focal point.
- Inverted: The image will be upside down compared to the object.
- Smaller: The size of the image can be calculated using the magnification formula:
Magnification (m) = -v/u
Substituting the values:
m = -10.8 / -27 = 0.4
This means the image is 0.4 times the size of the object. Given that the candle is 2.5 cm tall, the height of the image will be:
Height of image = 0.4 * 2.5 cm = 1 cm
Effect of Moving the Candle Closer
If the candle is moved closer to the mirror, the object distance (u) becomes less than 27 cm. As the object approaches the focal point, the image distance (v) will increase. In this case, the screen must be moved further away from the mirror to maintain a sharp image. If the candle moves inside the focal length (less than 18 cm), the image will become virtual, upright, and larger, and the screen will no longer be able to capture the image since it will form behind the mirror.
In summary, for the initial setup, place the screen at 10.8 cm from the mirror to get a sharp, inverted, and smaller image of the candle. If the candle is moved closer, adjust the screen further away to accommodate the changes in image distance.
