A convex mirror of focal length 30cm.produces a real and inverted image of magnification 0.5. By how much distance the object should be brought closer to obtain a virtual and erect image of magnification 2?
To answer this question, we need to use the mirror formula and the magnification formula.
The mirror formula is given by:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance from the mirror
u = object distance from the mirror
The magnification formula is given by:
magnification = -v/u
Given:
Focal length, f = 30 cm
Magnification, m = 0.5
We can use the magnification formula to find the image distance when the object distance is known:
0.5 = -v/u
Now, we need to find the object distance (u) corresponding to the given magnification.
Let's solve for v first:
0.5 = -v/u
v = -0.5u
Now, we can substitute this value of v in the mirror formula:
1/30 = 1/(-0.5u) - 1/u
To simplify the equation, let's take the reciprocal of both sides:
30 = -0.5u - u
30 = -1.5u
Now, we can solve for u:
u = -30 / -1.5
u = 20 cm
So, the object distance (u) required to obtain a real and inverted image with magnification 0.5 is 20 cm.
To find the new object distance (u') for a virtual and erect image with magnification 2, we can use the magnification formula again:
2 = -v/u'
Now, we need to find the new image distance (v') when the magnification and object distance are known.
Let's solve for v' first:
2 = -v'/u'
Now, we can substitute the known value of v (from the previous calculation) and solve for u':
2 = -(-0.5u')/u'
2 = 0.5
u' = -0.5/2
u' = -0.25
Since distances cannot be negative, we can disregard the negative sign and take the magnitude:
u' = 0.25
Therefore, to obtain a virtual and erect image with a magnification of 2, the object should be brought closer by a distance of 0.25 cm (or 2.5 mm).
To find the distance by which the object should be brought closer to obtain a virtual and erect image of magnification 2, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f = focal length of the convex mirror (30 cm)
v = image distance
u = object distance
We are given that the magnification for the real and inverted image is 0.5, and we want to find the object distance for a virtual and erect image with a magnification of 2. Let's denote this object distance as u'.
Given that magnification (m) is given by:
m = -v/u
For the real and inverted image:
m = 0.5
v = -0.5u
For the virtual and erect image:
m = 2
v' = -2u'
Now, let's substitute the values into the mirror formula for the real and inverted image:
1/30 = 1/(-0.5u) - 1/u
Multiply through by 30u:
u = -0.5u + 30
Simplifying the equation:
1.5u = 30
u = 20 cm
Now, let's substitute the values into the mirror formula for the virtual and erect image:
1/30 = 1/(-2u') - 1/u'
Multiply through by 30u':
u' = -2u' + 30
Simplifying the equation:
3u' = 30
u' = 10 cm
Therefore, the object should be brought 10 cm closer to the convex mirror to obtain a virtual and erect image of magnification 2.