A spherical mirror is polished on both sides. When the concave side is used as a mirror, the magnification is +1.2. What is the magnification when the convex side is used as a mirror, the object remaining the same distance from the mirror?
Well, when a mirror says it's polished on both sides, it's basically saying, "Hey, I'm a mirror, and I'm ready to reflect some light!" But enough about mirror small talk, let's get to the magnification business.
When the concave side is used as a mirror and the magnification is +1.2, it means the image is magnified and right-side up. So, maybe your reflection will make you look like a superhero or a noble steed. Who knows?
Now, when we flip things over and use the convex side as a mirror, it's like saying, "Okay, mirror, show us your other side!" This time, the image might be a bit smaller because convex mirrors tend to create smaller, virtual images. It's like looking at yourself in a funhouse mirror -- you might look smaller, taller, or maybe even squished!
But let's get back to the point: the magnification. Since the object is the same distance from the mirror, we can expect the image to be smaller, probably with a magnification less than +1.2. Unfortunately, without specific information or measurements, I can't give you an exact magnification value. However, it's safe to say that the magnification will be less than +1.2 and the image will be smaller compared to using the concave side.
So, there you have it! Flipping the mirror over to the convex side might shrink your marvelous magnified image from the concave side. Time to practice your tiny dance moves or deliver a speech with a chipmunk voice!
To find the magnification when the convex side is used as a mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
f = focal length of the mirror
v = image distance
u = object distance
Since we are told that the object remains the same distance from the mirror, the object distance (u) will be the same as before. Let's assume it is denoted as u1.
Now, since the mirror is polished on both sides, the radius of curvature (R) of the concave side will be equal to the radius of curvature of the convex side. Let's assume this as R.
For the concave side:
The magnification (m1) is given as +1.2, which means the image is upright. Therefore, the image distance (v1) will be positive.
Using the mirror formula, we can substitute the values:
1/f = 1/v1 - 1/u1
As per the sign convention for spherical mirrors, the focal length will be positive for a concave mirror.
We know that the focal length (f) = R/2.
Therefore:
1/(R/2) = 1/v1 - 1/u1
2/R = 1/v1 - 1/u1
(2u1 - 2v1) / (u1v1) = 1/R
Now, let's find the magnification for the convex side.
For the convex side:
The object distance (u2) is the same as u1.
Let's assume the image distance as v2, and the magnification as m2.
Using the mirror formula:
1/f = 1/v2 - 1/u2
Since the focal length (f) = R/2:
1/(R/2) = 1/v2 - 1/u2
Using the same logic as before, we can rewrite this equation as:
(2u2 - 2v2) / (u2v2) = 1/R
Now, we have two equations:
(2u1 - 2v1) / (u1v1) = 1/R ----(1)
(2u2 - 2v2) / (u2v2) = 1/R ----(2)
Since the radius of curvature (R) is the same for both sides, we can equate equations (1) and (2):
(2u1 - 2v1) / (u1v1) = (2u2 - 2v2) / (u2v2)
Canceling the common terms and rearranging the equation, we get:
u1/u2 - v1/v2 = 1
Now, since the object distance (u) is the same for both sides (u1 = u2), we can simplify the equation to:
1 - v1/v2 = 1
Simplifying further, we get:
v1/v2 = 0
Therefore, the magnification (m2) when the convex side is used as a mirror is 0.
To determine the magnification when the convex side of the spherical mirror is used, we need to use the mirror formula:
1/f = 1/v - 1/u,
where:
- f is the focal length of the mirror,
- v is the image distance,
- u is the object distance.
Given that the magnification (m) is defined as the ratio of the height of the image (h') to the height of the object (h), we can use the equation:
m = -v/u.
We are also given that when the concave side is used as a mirror, the magnification (m) is +1.2. Since concave mirrors have positive focal lengths, we can assume the magnification on the concave side to be positive as well. Thus, we have:
m_concave = +1.2 = -v/u.
When the concave side is used, the formula for magnification becomes:
m_concave = +1.2 = -v_concave/u.
Now, let's find the magnification (m_convex) when the convex side is used as a mirror while the object remains the same distance from the mirror.
When using the convex side of the mirror, the radius of curvature is reversed compared to the concave side, resulting in a negative focal length. Let's assume the negative focal length as "-f". Therefore, the formula for magnification on the convex side becomes:
m_convex = -v_convex/u.
Since the object distance remains the same, i.e., u = u_concave = u_convex, we can equate the magnifications:
m_concave = m_convex.
Substituting the given value of m_concave = +1.2, we have:
1.2 = -v_convex/u.
Now, we need to find the image distance (v_convex) using the mirror formula:
1/f = 1/v_convex - 1/u,
Substituting the given value of f as "-f", we get:
1/(-f) = 1/v_convex - 1/u.
Simplifying the equation, we have:
-1/f = 1/v_convex - 1/u.
Since the object distance (u) remains the same, we can substitute u for 1/u in the equation. This gives us:
-1/f = 1/v_convex - u/u.
Since u/u = 1, the equation simplifies to:
-1/f = 1/v_convex - 1.
Rearranging the equation, we have:
1/v_convex = 1/f - 1.
Finally, substituting the magnification equation (m_convex = -v_convex/u) into the equation, we get:
1.2 = -(1/f - 1)/u.
Simplifying the equation further, we have:
1.2 = -1/fu + 1/u.
Combining the terms, we have:
1.2 = (1 - fu)/u.
Multiplying both sides by "u", we get:
1.2u = 1 - fu.
Bringing all terms to one side, we have:
fu + 1.2u - 1 = 0.
This is a quadratic equation in terms of "u". We can solve for "u" using the quadratic formula:
u = (-b ± √(b^2 - 4ac))/(2a),
where a = f, b = 1.2, and c = -1.
Once we find the value of "u", we can substitute it into the magnification equation (m_convex = -v_convex/u) to determine the magnification when the convex side is used as a mirror.