a mirror which is somewhere between the object and image, forms a virtual upright image of the object 45 cm way from the object, measured on the optical axis of the mirror. The image is the size of the object. What kind of mirror must be used to produce this image. How far from the object must the mirror be placed. What are the focal lenght and the radius of curvature of the mirror.
To solve this problem, we can use the mirror equation and the magnification formula. Let's break down the information given step by step:
1. A virtual upright image is formed: This means that the image is formed behind the mirror, and it appears upright, the same way the object is oriented.
2. The image is the same size as the object: This implies that the magnification (m) is equal to 1.
3. The image is formed 45 cm away from the object, measured on the optical axis of the mirror: This is the distance between the object and the image. Let's call this distance d.
Based on this information, we can determine the type of mirror and the required placement distance.
1. Virtual upright image: This implies that the mirror must be a concave mirror. Concave mirrors can produce both real and virtual images, depending on the distance between the object and the mirror.
2. The image is the same size as the object (m = 1): For a concave mirror, the magnification formula is given by m = -(di/do), where di is the image distance and do is the object distance. Since we have m = 1, we can rewrite the formula as 1 = -(di/do).
3. The image is formed 45 cm away from the object: This means that di = -45 cm.
Using the magnification formula, we can substitute the given values and solve for do:
1 = -(di/do)
1 = -(-45/do)
1 = 45/do
Cross-multiplying, we get do = 45 cm.
Therefore, the mirror needs to be placed 45 cm away from the object.
Now, let's determine the focal length and radius of curvature of the mirror.
The focal length (f) of a concave mirror is half of the radius of curvature (R). The mirror equation for a concave mirror is given by 1/f = 1/do + 1/di.
Given that di = -45 cm and do = 45 cm, we can substitute these values into the mirror equation:
1/f = 1/45 + 1/-45
1/f = (-1 + 1)/45
1/f = 0/45
1/f = 0
Since 1/f = 0, this implies that the focal length (f) is infinite, which means the radius of curvature (R) is also infinite.
In conclusion:
- The mirror must be a concave mirror.
- The mirror must be placed 45 cm away from the object on the optical axis.
- The focal length of the mirror is infinite, and therefore, the radius of curvature of the mirror is also infinite.