Question

A candle flame 1.6 cm high is imaged in a ball bearing of diameter 0.4 cm. If the ball bearing is 20 cm away from the flame, find the location and the height of the image. (Assume the paraxial approximation to hold)

• A. 1 mm behind the mirror and 0.08 mm high
• B. 1 mm in front of the mirror and 0.08 mm high
• C. 1 mm behind the mirror and 0.8 mm high
• D. 10 mm in front of the mirror and 0.08 mm high

Answer 1

The correct option is A 1 mm behind the mirror and 0.08 mm high

The ball bearing surface will act as a convex mirror, its diameter,$D=0.4cm,so,f=\frac{D}{4}=0.1cm$
Also, according to the sign convention,
u=−20 cm.
Now applying mirror formula,
$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$
Or,
$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}$.
Substituting the values;
$\frac{1}{v}=\frac{1}{0.1}-\frac{1}{-20}=\frac{20+0.1}{2}$,
We get,
$\frac{1}{v}=10.05c{m}^{-1}$.
Hence, v=+0.1 cm or 1 mm behind the mirror, or oposite to the side the object (A positive value of v means image lies behind the mirror).
Now, for image height:
$\frac{h_i}{h_0}=-\frac{v}{u}$,
or,$\frac{h_i}{+1.6}=-\frac{+0.1}{-20}$
Therefore, hi=0.08 mm, is the height of the image.
Also, it is positive meaning that the image is errect, and the image is diminished, as expected!