You look at yourself in a shiny 8.4-cm-diameter Christmas tree ball. If your face is 25.0 cm away from the ball’s front surface, where is your image? Is it real or virtual? Is it upright or inverted?

검증된 단계별 안내

Determine the radius of curvature (R) of the spherical mirror. Since the diameter of the ball is 8.4 cm, the radius is half of that: R = 8.4 cm / 2 = 4.2 cm. The focal length (f) of a spherical mirror is related to the radius of curvature by the formula:

f = \frac{R}{2}

. Substitute R = 4.2 cm to find f.

Identify the type of mirror. A shiny Christmas tree ball is a convex mirror because the reflective surface is on the outside of the sphere. For convex mirrors, the focal length (f) is negative. Use the value of f calculated in the previous step, but assign it a negative sign.

Use the mirror equation to find the image distance (d_i):

\frac{1}{f} = \frac{1}{d}

_o+1d_i, where d_o is the object distance (25.0 cm), f is the focal length, and d_i is the image distance. Rearrange the equation to solve for d_i:

\frac{1}{d}

_i=1f-1d_o.

Substitute the known values of f (negative) and d_o (25.0 cm) into the rearranged mirror equation. Perform the subtraction to find the reciprocal of d_i, then take the reciprocal of that result to find d_i. Remember that for convex mirrors, the image distance will always be positive, indicating that the image is virtual.

Analyze the nature of the image. For convex mirrors, the image is always virtual (it cannot be projected onto a screen), upright (not inverted), and smaller than the object. Confirm these characteristics based on the calculated image distance and the properties of convex mirrors.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Concave Mirrors

The shiny surface of the Christmas tree ball acts like a concave mirror, which can form images based on the position of the object relative to its focal point. Concave mirrors can produce both real and virtual images depending on the object's distance from the mirror's surface.

Image Formation

The formation of an image by a mirror involves the principles of reflection and the mirror equation. The distance of the object from the mirror and the radius of curvature determine where the image is formed, whether it is real or virtual, and its orientation (upright or inverted).

Mirror Equation and Magnification

The mirror equation (1/f = 1/do + 1/di) relates the focal length (f), object distance (do), and image distance (di). Magnification (m) indicates the size and orientation of the image, calculated as m = -di/do. A positive magnification indicates an upright image, while a negative value indicates an inverted image.