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A convex mirror is placed on a stand, and an object is placed at a distance of $d=20 cm$ in front of it. The observer looks at the object’s image formed by the mirror, which appears to be at a distance $D=-15 cm$ behind the mirror. When the observer shifts $x=5 cm$ distance to the right, the apparent position of the image shifts by $X=3 cm$ distance to the left. Determine the focal length $f$ of the convex mirror using the parallax method.
Option: 1
$0.7 cm$

Option: 2
$1.2 cm$

Option: 3
$0.6 cm$

Option: 4
$2.4 cm$

Answers (1)

best_answer

Step 1: In the parallax method, the observer shifts and measures the apparent shift of the image when looking at it from different angles.
Given: Object distance $(d)=20 cm$ Apparent image distance $(D)=-15 cm$ Shift of observer $(x)=5 cm$ Shift of apparent image $(X)=3 cm$ Focal length of the convex mirror $(f)=f$
Step 2: Use the formula for the magnification $(m)$ of the mirror to relate the object and image distances.

$m=-\frac{D}{d}$

Step 3: Use the formula for magnification $(m)$ in terms of the focal length $(f)$ and object distance $(d)$ .

$m=-\frac{f}{d}$

Step 4: Equate the two expressions for magnification $(m)$ and solve for the focal length $(f)$ .

$-\frac{D}{d}=-\frac{f}{d}$

$D=f$

Step 5: Use the formula for parallax to relate the shift of observer $(x)$ , shift of apparent image $(X)$ , and the object and image distances.

$x=\frac{X}{D-f}$

Step 6: Substitute the value of $D$ from step 4 into the parallax formula.

$x=\frac{X}{f}$

Step 7: Solve for the focal length $(f)$ .

$f=\frac{X}{x}=\frac{3 cm}{5 cm}=0.6 cm$

So, the focal length $(f)$ of the convex mirror is $0.6 cm$ .
Therefore, the correct answer C.

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