Projecting a Magnified Real Image on a Fixed Screen
A student is asked to project a real, magnified image of a 2.0 cm tall candle on a screen using a convex lens of focal length 15.0 cm. The screen is fixed 80.0 cm from the lens, and the available screen height is 10.0 cm. The student first places the candle 10.0 cm from the lens and finds no image on the screen. The student then moves the candle until a sharp image appears. The lens has a diameter of 4.0 cm, and the room is darkened. The examiner asks the student to locate the object, check whether the image fits the screen, and explain why the first attempt failed. Distances are measured from the lens along the principal axis, and the lens is thin. The candle is aligned with the principal axis.
1. Find the object distance that places the image on the screen at 80.0 cm. Use the lens formula and state the sign convention.
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Take the lens as origin and incident light from the left, so v = +80.0 cm and f = +15.0 cm.
The thin-lens formula 1/f = 1/v - 1/u gives1/u = 1/80 - 1/15 = (15 - 80)/1200 = -65/1200,
so u = -18.46 cm.
Thus the candle must be 18.5 cm to the left of the lens. This lies between f = 15 cm and 2f = 30 cm, which is the condition for a real, magnified image. With the screen fixed at 80 cm, there is only this one object position for a sharp real image.
2. Calculate the magnification and image height. Is the available 10.0 cm screen height sufficient? Describe the orientation.
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m = v/u = 80.0 / (-18.46) = -4.33.
The negative sign shows that the image is inverted.
Image height = |m| × object height = 4.33 × 2.0 = 8.66 cm.
This is less than 10.0 cm, so the image fits on the screen with about 1.3 cm total margin. The actual image height is 8.7 cm, and it is inverted relative to the candle.
3. Explain why placing the candle at 10.0 cm produced no screen image.
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At u = -10.0 cm the object lies inside the focal length because 10 cm < 15 cm. The rays after the lens diverge, so the lens forms a virtual, upright, magnified image on the same side as the object. A virtual image cannot be caught on a screen. The student must move the candle beyond 15 cm from the lens; for a screen at 80 cm, the required position is 18.5 cm, as calculated.