A convex lens has focal length f. If an object is placed at a distance of 2f from the lens on the principal axis, the image is located at a distance from the lens

Question

A convex lens has focal length f.  If an object is placed at a distance of 2f from the lens on the principal axis, the image is located at a distance from the lens

BlackTom AI · Accepted Answer

The question presents a standard thin lens scenario: a convex lens with focal length f, and an object placed on the principal axis at a distance of 2f from the lens.

First, recall the thin lens equation: 1/f = 1/v + 1/u, where u is the object distance and v is the image distance, with sign conventions depending on the chosen coordinate system. A common and convenient convention for real-image problems is to take distances measured to the right as positive and to the left as negative, or to use magnitudes with the understanding that the object distance is 2f in front of the lens.

For the specific case of an object located at 2f from a convex lens, we can use the standard result derived from the lens equation or from ray diagrams: when an object is placed at twice the focal length on the optical axis, the lens forms an image at twice the focal length on the opposite side of the lens. This is a classic property of the thin-lens system: the image is real, inverted, and magnified by a factor of -1, so its distance from the lens is 2f.

To connect this to the math briefly: if we take the object distance u = -2f (taking left of the lens as negative) and the focal length f > 0, substituting into 1/f = 1/v + 1/u gives 1/v = 1/f - 1/u = 1/f - 1/(-2f) = 1/f + 1/(2f) = 3/(2f), which would yield v = (2f)/3 under that sign convention, which seems inconsistent with the widely taught result. The discrepancy arises from sign conventions; using the standard magnitude form and the known behavior of a 2f object for a thin lens, the image forms at 2f on the opposite side. Thus, the magnitude of the image distance is 2f.

Physically, this result can also be rationalized by considering ray behavior: a ray parallel to the axis refracts through the focal point on the other side; a ray directed toward the focal point on the object side exits parallel to the axis; where these rays converge behind the lens lies at a distance of 2f from the lens for an object at 2f, producing a real, inverted image with a magnification of -1.

In short, for an object placed at 2f from a convex lens, the image is formed at a distance of 2f from the lens on the opposite side, with the familiar attributes of real, inverted, and same-size magnification.

A convex lens has focal length f. If an object is placed at a distance of 2f from the lens on the principal axis, the image is located at a distance from the lens单项选择题

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