A convex mirror used as a security mirror in a store has a radius of curvature of 20\text{… — JAMB UTME

A convex mirror used as a security mirror in a store has a radius of curvature of $20\text{ cm}$ . If a shopper stands $30\text{ cm}$ in front of the mirror, at what distance from the mirror is the shopper's image formed?

$7.5\text{ cm}$ behind the mirrorAnswer
B
$15.0\text{ cm}$ in front of the mirror
C
$12.0\text{ cm}$ behind the mirror
D
$15.0\text{ cm}$ behind the mirror

Answer

The shopper's image is formed

7.5\text{ cm}

behind the mirror.

For a convex mirror, the focal length is negative and given by

f = -r/2 = -10\text{ cm}

. Substituting

f = -10\text{ cm}

and object distance

u = +30\text{ cm}

into the mirror equation

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

gives

\frac{1}{v} = -\frac{1}{10} - \frac{1}{30} = -\frac{4}{30}

, which solves to

v = -7.5\text{ cm}

. A negative image distance represents a virtual image formed

7.5\text{ cm}

behind the mirror.

Step-by-Step Solution

Determine the focal length of the convex mirror using its radius of curvature

f = -\frac{r}{2} = -\frac{20\text{ cm}}{2} = -10\text{ cm}

By sign convention, a convex mirror has a negative focal length equal to half its radius of curvature.

Apply the mirror formula to find the image distance

v

\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \implies \frac{1}{-10} = \frac{1}{30} + \frac{1}{v}

The mirror formula relates focal length (

f

), object distance (

u

), and image distance (

v

Solve for

\frac{1}{v}

and calculate

v

\frac{1}{v} = -\frac{1}{10} - \frac{1}{30} = -\frac{3 + 1}{30} = -\frac{4}{30} = -\frac{2}{15}\text{ cm}^{-1} \implies v = -7.5\text{ cm}

The negative sign indicates that the image is virtual and located behind the mirror.

Key Concept

Mirror Formula and Sign Convention for Convex Mirrors

Estimated Time:1m 30s

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