How Does Magnification Change When Switching from Convex to Concave Mirror?

[gamesandmore]

A spherical mirror is polished on both sides. When used as a convex mirror, the magnification
is +1/4. What is the magnification when used as a concave mirror, the object remaining the
same distance from the mirror?

I started with:

mconvex = 1/4
mconcave = ?

doconvex = doconcave
hoconvex = hoconcave

I wasn't sure about how to go from here...:

m = hi/ho = -di/do

mconvex = hiconvex/hoconvex = -diconvex/doconvex
mconcave = hiconcave/hoconcave = -diconcave/doconcave

1/4 = hiconvex/hoconvex
hoconvex = 4hiconvex

mconcave = hiconcave / hoconcave
hoconcave = hiconcave / mconcave

4hiconvex = hiconcave / mconcave

mconcave = hiconcave / 4hiconvex

I wasn't sure what to do from here or if I approached this correctly

 

[mgb_phys]

This probably does a better job of drawing the diagrams than I could.
https://www.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/mirror-calcs.htm

Basically it's just a matter of getting the signs for the focal length correct.

 

[ogb p]

.

Your approach is correct. To find the magnification when used as a concave mirror, we can use the relationship between object and image distances for a spherical mirror:

1/do + 1/di = 1/f

Where:
do = object distance
di = image distance
f = focal length

Since the object distance remains the same, we can rewrite the equation as:

1/di = 1/f - 1/do

For a convex mirror, the focal length is negative, so we can rewrite as:

1/di = -1/|f| - 1/do

And since we know that for a convex mirror, the magnification is +1/4, we can substitute that in:

1/di = -1/|f| - 1/do = +1/4

Solving for di, we get:

di = 4do/(3do - |f|)

Now, for a concave mirror, we can use the same equation and substitute in the focal length for a concave mirror, which is positive:

1/di = 1/f - 1/do = 1/|f| - 1/do

And since we want to find the magnification, we can substitute in the image distance we just found for a convex mirror:

1/di = 1/|f| - 1/do = 1/|f| - 1/(4do/(3do - |f|))

Solving for di, we get:

di = 3do/(3do - |f|)

And using the magnification formula, we can find the magnification for a concave mirror:

mconcave = di/do = (3do/(3do - |f|))/do = 3/(3 - |f|/do)

Since we know that do = ho (the object height) and di = hi (the image height), we can rewrite the equation as:

mconcave = hi/ho = 3/(3 - |f|/ho)

So, the magnification when used as a concave mirror, with the object remaining the same distance from the mirror, is 3/(3 - |f|/ho). This value will vary depending on the specific values of the object distance and the focal length of the concave mirror.