Focal length of a convex mirror is 10 cm where should an object be placed in front of a convex mirror so that the image of half the size of the object is formed?
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If you combine the formula relating object and image distance with the magnification formula, then solve for object distance, you will see that the distance must be the same as the focal length to get a 1/2 sized image.
Maybe this web site will help.
https://www.quora.com/A-convex-mirror-of-focal-length-f-produces-an-image-of-1-nth-of-the-size-of-the-object-What-is-the-distance-of-the-object-from-the-mirror
To find the position where an object should be placed in front of a convex mirror to form an image of half the size, we need to use the mirror formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the convex mirror
- v is the image distance (distance of the image from the mirror)
- u is the object distance (distance of the object from the mirror)
Given:
f = 10 cm
We want the image size, magnification (m), to be half the object size. So, m = -1/2 (negative sign indicates a virtual image).
Using the magnification formula:
m = v/u
Given m = -1/2, we can substitute this value in the magnification formula:
-1/2 = v/u
Now, we can rearrange and solve for v:
v = -u/2
Substitute this value of v in the mirror formula:
1/f = 1/v - 1/u
1/10 = 1/(-u/2) - 1/u
Now, we can simplify the equation using common denominators:
1/10 = 2/(2u) - 2/2u
1/10 = (2 - 2)/2u
1/10 = 0/2u
Since the equation is satisfied for any value of u, there is no specific position where the image will be half the size of the object for a convex mirror.
Therefore, an image of half the size of the object cannot be formed by a convex mirror.
To determine where an object should be placed in front of a convex mirror to form an image of half its size, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the convex mirror
- v is the image distance (distance between the mirror and the image)
- u is the object distance (distance between the mirror and the object)
Given that the focal length (f) of the convex mirror is 10 cm, and we want the image to be half the size of the object, we can use the magnification formula:
m = -v/u
Where:
- m is the magnification
Since we want the image to be half the size of the object, m = -1/2.
Now, let's substitute the given values into the equations:
1/10 = 1/v - 1/u
-1/2 = -v/u
To solve these equations, let's take the reciprocal on both sides of the second equation:
1/(-1/2) = -u/v
-2 = -u/v
Now, substitute this value of -u/v into the first equation:
1/10 = 1/v - 1/(-2)
Simplify the equation:
1/10 = 1/v + 1/2
Find a common denominator:
1/10 = (2 + v) / (2v)
Now, cross-multiply:
2v + v = 20
3v = 20
v = 20/3 ≈ 6.67 cm
So, the image should be approximately 6.67 cm away from the convex mirror.
Remember, when solving these types of problems, it's essential to keep track of the sign conventions. In this case, negative (-) denotes that the image is formed on the same side as the object (virtual image).