Question

How do you find the position and magnification of a convex mirror?

Assume the reflected object is `3.00` cm high and is placed `20.0` cm from the convex mirror with focal length of `8.00` cm.

Answer 1

See below.
Position: `-0.05714` `"m"`
Magnification: `0.28571`

Explanation:

Since we are given that the height of the object is `0.03` meters high and is placed `0.2` meters away from a convex mirror with focal length `-0.08` meters, we can write out are givens in SI units as:
`d_(obj)=.2`
`h_(obj)=.03` since the image created by convex mirrors are always upright and thus have a positive height value
`f=-0.08` since the focal length of convex mirrors are negative

Consider the following two formulas:

Lensmaker's Formula:
`1/f=1/d_(obj)+1/d_(img)`

Magnification Equation:
`M=h_(img)/h_(obj)=-d_(img)/d_(obj)`

To determine the image's position, we can solve for `d_(img)` in the Lensmaker's Formula with variables only, then plug in the given values to solve:
`1/d_(img)=1/f-1/d_(obj)`
Taking the reciprocal of both sides, we get:
`d_(img)=1/(1/f-1/d_(obj)`

Now, we can substitute the given values of `d_(obj)` and `f` to solve:
`d_(img)=1/(1/-0.08-1/0.2)`
`=-0.05714` `"m"`

Using this value, we can find the magnification of the convex mirror:
`M=-d_(img)/d_(obj)`
`=-(-0.05714)/0.2`
`=0.28571` which has no units

Answer 2

See below.

Explanation:

You will need to calculate the distance between the mirror and the image first, which can be done using the mirror equation:

`1/f=1/d_(o)+1/d_i`

where `f` is the focal length, `d_o` is the distance between the mirror and the object, and `d_i` is the distance between the mirror and the image.

We can solve for `d_i`:

`=>1/d_i=1/f-1/d_(o)`

`=>d_i=(1/f-1/d_(o))^-1`

Note that because this is a convex mirror, the focal length must be negative.

Given that `d_o=20.0cm` and `f=-8.00cm` :

`d_i=(-1/8-1/20)^-1`

`=(-7/40)^-1`

`=-40/7cm`

The magnification of a curved mirror can be expressed by the following equation:

`m=-d_i/d_(o)`

Thus we have:

`m=(-(-40/7))/20`

`m=40/140=2/7`

`:.` The position of the image is `40/7` cm behind the mirror and the magnification of the mirror is `2/7`.

This answer makes sense, as a convex mirror will always produce an image which is reduced, upright, and virtual.