An object whose height is 3.5 cm is at a distance of 8.5 cm from a spherical concave mirror. Its image is real and has a height of 11.2 cm. Calculate the radius of curvature of the mirror.
How far from the mirror is it necessary to place the above object in order to have a virtual image with a height of 11.2 cm?
You should be familiar with the lens and mirror focusing equation. Here's a chance to learn them and try them out.
For a real image formed by a concave mirror:
1/Do + 1/Di = 1/f = 2/R
You don't need to height of the object to compute the radius, R.
Di = 11.2 and Do = 8.5.
To have a magnification of 11.2/3.5 = 3.2, you must have Di/Do = 2.2
Since you now know R, you can use the mirror equation to solve for the new object distance Do
To calculate the radius of curvature of the spherical concave mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the mirror
- v is the distance of the image from the mirror
- u is the distance of the object from the mirror
We are given the following information:
- Height of the object (h₁) = 3.5 cm
- Distance of the object from the mirror (u₁) = 8.5 cm
- Height of the image (h₂) = 11.2 cm
Let's calculate the radius of curvature of the mirror first:
Since the image is real and inverted, the height of the image is negative. Therefore, h₂ = -11.2 cm.
Using the magnification formula:
magnification (m) = h₂ / h₁
m = -11.2 cm / 3.5 cm
m ≈ -3.2
Since m = -v/u, we can rearrange the equation to get:
v = -m * u
Substituting the values:
v = -(-3.2) * 8.5 cm
v ≈ 27.2 cm
Now, let's calculate the radius of curvature (R):
R = 2 * f
Using the mirror formula, we know that:
1/f = 1/v - 1/u
Substituting the values:
1/f = 1/27.2 cm - 1/8.5 cm
1/f ≈ 0.0368 cm^(-1)
Therefore:
f ≈ 1 / 0.0368 cm^(-1)
f ≈ 27.2 cm
Now, let's determine the distance of the object from the mirror (u₂) to have a virtual image with a height of 11.2 cm:
Using the magnification formula:
m = -h₂ / h₁
-3.2 = 11.2 cm / u₂
Rearranging the equation:
u₂ = 11.2 cm / -3.2
u₂ ≈ -3.5 cm
The negative sign indicates that the object needs to be placed on the same side as the virtual image, behind the mirror.
Therefore, the object needs to be placed approximately 3.5 cm from the mirror to have a virtual image with a height of 11.2 cm.