A concave spherical mirror has a focal length of 20 cm. An object is placed 10 cm in front of the mirror on the mirror's axis. Where is the image located?

Question

A concave spherical mirror has a focal length of 20 cm.  An object is placed 10 cm in front of the mirror on the mirror's axis.  Where is the image located?

BlackTom AI · Accepted Answer

Consider the setup: a concave spherical mirror with focal length f = +20 cm and an object placed at u = -10 cm (taking the standard Cartesian convention where distances in front of the mirror are negative).

Apply the mirror equation 1/f = 1/v + 1/u to find the image distance v.

- Substitute the known values: 1/20 = 1/v + 1/(-10).
- This gives 1/v = 1/20 + 1/10 = 0.05 + 0.10 = 0.15.
- Therefore v = 1/0.15 ≈ 6.67 cm.
- The positive value of v indicates the image lies on the same side as the object, i.e., in front of the mirror.

Now evaluate each option:

Option A: '20 cm in front of the mirror' — This would imply v ≈ 20 cm, which does not satisfy the mirror equation with u = -10 cm and f = +20 cm. The calculated image distance is about 6.7 cm, not 20 cm, so this is incorrect.

Option B: '6.7 cm behind the mirror' — 'Behind the mirror' would correspond to a negative v in the standard sign convention. Since our calculation yields v ≈ +6.7 cm, this option misplaces the image and is therefore incorrect.

Option C: '6.7 cm in front of the mirror' — This matches the computed v ≈ +6.7 cm, indicating the image is indeed formed 6.7 cm in front of the mirror on the same side as the object. This aligns with the sign convention and the mirror equation, so this statement is correct.

Option D: '20 cm behind the mirror' — As with option B, placing the image behind the mirror would require a negative v of about -20 cm, which contradicts the calculation from 1/f = 1/v + 1/u. Hence this is incorrect.

In summary, only option C satisfies the optical equation with the given f and u, and places the image on the correct side of the mirror. The other choices either contradict the computed value or misinterpret the image’s location with respect to the mirror’s front or back side.

A concave spherical mirror has a focal length of 20 cm. An object is placed 10 cm in front of the mirror on the mirror's axis. Where is the image located?Single choice

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