A lamp is 4m from wall. Calculate the focal length of a concave mirror which forms a five time magnified image of lamp on wall. How far from wall must the mirror be placed?
To find the focal length of the concave mirror, we can use the magnification formula:
Magnification (m) = - (image distance / object distance)
Here, the magnification is given as 5, and the object distance is the distance of the lamp from the mirror, which is 4m. We need to find the image distance.
Since the magnification is positive (+5), we know that the image formed by the concave mirror is virtual and magnified.
Now, let's calculate the image distance using the magnification formula:
5 = - (image distance / 4)
Cross-multiplying the equation:
-5 * 4 = image distance
Image distance = -20m
The negative sign indicates that the image is formed on the same side as the object (virtual image).
Now, to find the distance of the mirror from the wall, we need to consider the mirror equation:
1/f = 1/di + 1/do
Where:
f = focal length of the mirror
di = image distance (-20m)
do = object distance (4m)
Solving the equation for f:
1/f = 1/(-20) + 1/4
1/f = -1/20 + 1/4
1/f = (-1 + 5) / 20
1/f = 4/20
1/f = 1/5
f = 5m
Therefore, the focal length of the concave mirror is 5m.
To find the distance from the wall at which the mirror must be placed, it is equal to the focal length of the mirror (5m), since the image will be formed at infinity as the magnification is 5.