an object of 1 cm^2 face area is placed at a distance of 1.5m from a screen. How far from the object should a concave mirror be placed si that it forms 4 cn^2 image on the screen?
To find the distance from the object at which the concave mirror should be placed to form a 4 cm^2 image on the screen, we can use the magnification formula:
magnification = image size / object size
Given that the object's face area is 1 cm^2 and the image size on the screen is 4 cm^2, we have:
magnification = 4 cm^2 / 1 cm^2 = 4
The magnification of the concave mirror is given by the formula:
magnification = -image distance / object distance
We are given that the object is placed at a distance of 1.5 m from the screen. Let's denote this distance as "d_object".
magnification = -image distance / d_object
Since we know the magnification value is 4, we can rearrange the formula to solve for the image distance:
image distance = -4 * d_object
Now, we need to find the object distance for the concave mirror. The object distance is the distance between the object and the mirror. It can be denoted as "d_mirror".
Using the mirror formula:
1 / d_object + 1 / d_mirror = 1 / focal length
Let's assume the focal length of the concave mirror is "f".
1 / 1.5 + 1 / d_mirror = 1 / f
Simplifying the equation:
2 / d_mirror = 1 / f
Now, we can substitute the value of d_mirror in terms of f from the magnification formula:
d_mirror = -4 * d_object
Substituting this into the equation above:
2 / (-4 * d_object) = 1 / f
Simplifying:
1 / (-2 * d_object) = 1 / f
The denominator on the left side of the equation represents the object distance for the concave mirror (d_mirror), which we need to find.
Therefore, the concave mirror should be placed at a distance equal to the object distance, which is given by:
d_mirror = -2 * d_object
Substituting d_object = 1.5 m:
d_mirror = -2 * 1.5 m = -3 m
Since distance cannot be negative, we disregard the negative sign and take the magnitude, which gives the actual distance as 3 meters.
Therefore, the concave mirror should be placed 3 meters away from the object to form a 4 cm^2 image on the screen.
To determine the distance from the object to the concave mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance from the mirror (positive for real images)
u = object distance from the mirror (positive if the object is in front of the mirror)
Given:
Object face area = 1 cm^2
Object distance, u = 1.5 m
Image face area = 4 cm^2
We need to find the image distance, v, so we can find the focal length, f. The focal length of the mirror is the distance from the mirror to its focal point.
First, let's calculate the magnification, M:
M = (Image area / Object area) = (4 cm^2) / (1 cm^2) = 4
Since the magnification, M = -v/u, we can solve for v using the magnification formula:
v = -M × u
v = -4 × 1.5 m
v = -6 m (the negative sign indicates a real image)
Now, we can substitute the values of v and u into the mirror formula:
1/f = 1/v - 1/u
1/f = 1/(-6 m) - 1/(1.5 m)
1/f = -1/6 m - 2/3 m
1/f = (-1 - 4) / 6 m
1/f = -5 / 6 m
f = -6 / 5 m
The negative sign of the focal length indicates a concave mirror. The focal length, f, is approximately -1.2 m.
Therefore, the concave mirror should be placed approximately 1.2 meters away from the object to form a 4 cm^2 image on the screen.
size is proportionalto distance sqkuared
4/1=(distance/1.5)^2