How would you use a concave mirror of radius of convatrue 18cm to give a four time magnified erect image of an object?
To use a concave mirror to create a four times magnified erect image, you need to determine the position of the object and the focal length of the mirror.
Step 1: Given the radius of curvature (R) of the concave mirror is +18 cm. Since it is a concave mirror, R is positive.
Step 2: Use the mirror formula, which is given by:
1/f = 1/v - 1/u
where:
f - focal length of the mirror
v - image distance from the mirror (positive for a real image)
u - object distance from the mirror (positive for an object on the same side as the incident light)
Step 3: Let's assume the magnification factor (m) is 4, which means the image height (h') is four times the object height (h).
m = h'/h = -v/u
Since the image is erect, the magnification factor is positive.
Step 4: Rearrange the magnification formula to get the relationship between v and u:
v = -4u
Step 5: Substitute the value of v in terms of u into the mirror formula:
1/f = 1/-4u - 1/u
Step 6: Simplify the equation:
1/f = -5/u
Rearranging, we get:
f = -u/5
Step 7: Now, equate the focal length (f) given in terms of R (radius of curvature):
f = R/2
-u/5 = R/2
Solving for u, we get:
u = -5R/2
Substitute the value of R:
u = -5(18)/2 = -45 cm (negative sign indicates that the object is placed on the same side as the incident light)
Step 8: Now that we have the object distance, we can find the image distance using the magnification formula:
m = -v/u
4 = -v/-45
Solving for v, we get:
v = -180 cm (negative sign indicates a real image)
Step 9: Calculate the focal length (f):
f = -u/5 = 45/5 = 9 cm
Therefore, to create a four times magnified erect image using a concave mirror with a radius of curvature of 18 cm, the object must be placed at a distance of 45 cm from the mirror, and the focal length of the mirror should be 9 cm.