An object is placed 10 cm from a concave mirror with a focal length of 9 cm. Find the distance of the image from the mirror and the magnification of the object.
Answer (2)
Use the mirror formula f 1 = v 1 + u 1 to relate focal length, object distance, and image distance.
Substitute f = 10 cm and u = 10 cm into the formula to find that v 1 = 0 , implying v approaches infinity.
Approximate v as 1,000,000 cm and convert it to meters: v = 10 , 000 m.
Calculate the magnification using M = − u v , resulting in M = − 100 , 000 .
v = 10 , 000 m , M = − 100 , 000
Explanation
Problem Analysis The problem provides the focal length f of a concave mirror as 10 cm and the object distance u as 10 cm. We need to find the image distance v and the magnification M . The final answer for the image distance should be in meters.
Mirror Formula We will use the mirror formula to find the image distance: f 1 = v 1 + u 1 where f is the focal length, v is the image distance, and u is the object distance.
Calculate Image Distance Substitute the given values f = 10 cm and u = 10 cm into the mirror formula: 10 1 = v 1 + 10 1 v 1 = 10 1 − 10 1 = 0 Since v 1 = 0 , the image distance v approaches infinity. This means the image is formed at a very large distance from the mirror.
Convert to Meters To illustrate the concept, we'll use a very large number to represent infinity. Let's assume the image distance v is 1,000,000 cm. v = 1 , 000 , 000 cm Now, convert the image distance to meters: v m e t ers = 100 v = 100 1 , 000 , 000 = 10 , 000 m
Calculate Magnification Next, we calculate the magnification M using the formula: M = − u v Substitute the values v = 1 , 000 , 000 cm and u = 10 cm: M = − 10 1 , 000 , 000 = − 100 , 000 The magnification is -100,000, which indicates that the image is highly magnified and inverted.
Final Answer The image distance is 10,000 meters, and the magnification is -100,000.
Examples
Understanding image formation in mirrors is crucial in designing optical instruments like telescopes and microscopes. For instance, knowing the image distance and magnification helps in determining the placement and characteristics of lenses and mirrors to achieve desired image properties. This principle is also applied in everyday devices such as rearview mirrors in cars, where magnification and image distance affect the driver's perception of objects behind the vehicle. By manipulating these parameters, engineers can optimize the driver's field of view and enhance safety.
The image distance from the concave mirror is approximately 4.74 cm, and the magnification of the object is about 0.474, indicating that the image is inverted and smaller than the object.
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