A small cup of green tea is positioned on the central axis of a spherical mirror. The lateral magnification of the cup is +0.250, and the distance between the mirror and its focal point is 2.00cm. (a) What is the distance between the mirror and the image it produces? (b) Is the focal length positive or negative? (c) Is the image real or virtual?

Magnification, m = +0.250

Focal length, f = 2.0cm.

From the magnification value, we can determine the type of mirror. Then we can use the mirror formula to find the object distance from the mirror and magnification relation to relate image distance and object distance.

Formula:

Mirror formula $\frac{1}{f}=\frac{1}{p}+\frac{1}{i}$

Magnification, $m=-\frac{i}{p}$

(a)

We have the lateral magnification of cup.

Magnification, m= +0.250

As magnification is positive, the mirror is convex type. Hence,thefocal length is negative.

Now usingthemirror formula, we have

$\frac{1}{p}-\frac{1}{0.250p}=\frac{1}{f}$

$\frac{1}{p}-\frac{4}{p}=\frac{1}{f}$

This equation gives

$$\begin{gathered} -\frac{3}{p}=\frac{1}{f} \\ =\frac{1}{-2.0} \\ =0.5c{m}^{-1} \\ p=\frac{3}{0.5}cm \end{gathered}$$

=6.0cm

Using magnification equation,

i= -mp

= -0.250 * 6.0cm

= - 1.50cm

(b)

We have from part (a) that the focal length is negative.

It is f = -2.0cm.

(c)

As the focal length is negative and the mirror is convex, the image is virtual.