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

Assume the reflected object is 3.003.00 cm high and is placed 20.020.0 cm from the convex mirror with focal length of 8.008.00 cm.

2 Answers
Y
May 13, 2017

See below.
Position: -0.05714−0.05714 "m"m
Magnification: 0.285710.28571

Explanation:

Since we are given that the height of the object is 0.030.03 meters high and is placed 0.20.2 meters away from a convex mirror with focal length -0.08−0.08 meters, we can write out are givens in SI units as:
d_(obj)=.2dobj=.2
h_(obj)=.03hobj=.03 since the image created by convex mirrors are always upright and thus have a positive height value
f=-0.08f=−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)1f=1dobj+1dimg

Magnification Equation:
M=h_(img)/h_(obj)=-d_(img)/d_(obj)M=himghobj=−dimgdobj

To determine the image's position, we can solve for d_(img)dimg in the Lensmaker's Formula with variables only, then plug in the given values to solve:
1/d_(img)=1/f-1/d_(obj)1dimg=1f−1dobj
Taking the reciprocal of both sides, we get:
d_(img)=1/(1/f-1/d_(obj)dimg=11f−1dobj

Now, we can substitute the given values of d_(obj)dobj and ff to solve:
d_(img)=1/(1/-0.08-1/0.2)dimg=11−0.08−10.2
=-0.05714=−0.05714 "m"m

Using this value, we can find the magnification of the convex mirror:
M=-d_(img)/d_(obj)M=−dimgdobj
=-(-0.05714)/0.2=−−0.057140.2
=0.28571=0.28571 which has no units

Morgan
May 13, 2017

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_i1f=1do+1di

where ff is the focal length, d_odo is the distance between the mirror and the object, and d_idi is the distance between the mirror and the image.

We can solve for d_idi:

=>1/d_i=1/f-1/d_(o)⇒1di=1f−1do

=>d_i=(1/f-1/d_(o))^-1⇒di=(1f−1do)−1

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

Given that d_o=20.0cmdo=20.0cm and f=-8.00cmf=−8.00cm :

d_i=(-1/8-1/20)^-1di=(−18−120)−1

=(-7/40)^-1=(−740)−1

=-40/7cm=−407cm

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

m=-d_i/d_(o)m=−dido

Thus we have:

m=(-(-40/7))/20m=−(−407)20

m=40/140=2/7m=40140=27

:. 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.

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