凹レンズによる像の形成に関する記事
凹レンズによる像の形成を理解するために、次の例題と解答を見てください。この場合、物体は凹レンズから一定の距離にあると仮定します。
次に、凹レンズによる像の形成、凹レンズからの像距離、および凹レンズによる像の倍率を図示します。
凹レンズ（発散レンズ）の焦点距離は20cmです。高さ10cmの物体がレンズの左側にあります。以下の条件を満たす場合、像距離、像の倍率、像の高さを求めなさい。
a) the object distance is smaller than the focal length of the concave lens
b) the object distance is the same as the focal length of the concave lens
c) the object distance is greater than the focal length of the concave lens
既知：
The focal length of the concave lens (f) = -20 cm
The focal length of the concave lens has a negative sign because the focal point of the concave lens is virtual, where the light beam does not pass through the focal point.
The height of object (h) = 10 cm
解決策：
物体距離は凹レンズの焦点距離よりも小さい（do > f）。
物体までの距離が5cm、10cm、15cmだと仮定します。
a) The focal length is -20 cm and if the object distance (do) = 5 cm
画像距離（di）
1/di = 1/f – 1/do = -1/20 – 1/5 = -1/20 – 4/20 = -5/20
di = -20/5 = -4 cm
The image distance sign, negative, means the image is virtual, where the beam of light does not pass through the image.
The image distance, 4 cm, smaller than the object distance, 5 cm.
画像の倍率（M）
M = -di / do = -(-4)/5 = 4/5 = 0.8
The magnification of image smaller than 1 means that the image is reduced, or the image size is smaller than the object size.
The height of image (h’)
The equation of the magnification of image :
M = h' / h
h' = M h = (0.8)(10 cm) = 8 cm
The height of image is sign positive, means that the image is upright.
b) The focal length is -20 cm and if the object distance (do) = 10 cm
画像距離（di）
1/di = 1/f – 1/do = -1/20 – 1/10 = -1/20 – 2/20 = -3/20
di = -20/3 = -6.7 cm
The image distance is negative, means that the image is virtual, where the beam of light does not pass through the image.
The image distance 6.7 cm, smaller than the object distance, 10 cm.
画像の倍率（M）
M = -di / do = -(-6.7)/10 = 6.7 / 10 = 0.67
The magnification of image is smaller than 1 means that the image is reduced, or the image size is smaller than the object size.
The height of image (h’)
画像の拡大率の式：
M = h'/h
h' = M h = (0.67)(10 cm) = 6.7 cm
The height of image is sign positive, means that the image is upright.
c) The focal length is -20 cm and the object distance (do) = 15 cm
画像距離（di）
1/di = 1/f – 1/do = -1/20 – 1/15 = -3/60 – 4/60 = -7/60
di = -60/7 = -8.6 cm
The image distance is negative, means that the image is virtual, where the beam of light does not pass through the image.
The image distance is 8.7 cm, smaller than the object distance, 15 cm.
画像の倍率（M）
M = -di / do = -(-8.6)/10 = 8.6 / 10 = 0.86
The magnification of image smaller than 1 means that the image size is smaller than the object size.
The height of image (h’)
画像の拡大率の式：
M = h'/h
h' = M h = (0.86)(10 cm) = 8.6 cm
The height of the image is positive, means that the image is upright.
The object distance is the same as the focal length of the concave lens (do = f)
焦点距離（f）＝物体距離（do）＝20cm。
画像距離（di）
1/di = 1/f – 1/do = -1/20 – 1/20 = -2/20 = -2/20
di = -20/2 = -10 cm
The image distance is negative, means that the image is virtual, where the beam of light does not pass through the image.
The image distance is 10 cm, smaller than the object distance, 20 cm.
画像の倍率（M）
M = -di / do = -(-10)/20 = 10/20 = 0.5
The magnification of image is smaller than 1, means that the image size is smaller than the object size.
The height of the image (h’)
The equation of the magnification of the image :
M = h'/h
h' = M h = (0.5)(10 cm) = 5 cm
The height of the image is positive, means that the image is upright.
物体距離は凹レンズの焦点距離よりも大きい（do > f）
Suppose the distance of objects is 30 cm and 40.
a) The focal length is -20 cm and if the object distance (do) = 30 cm
画像距離（di）
1/di = 1/f – 1/do = -1/20 – 1/30 = -3/60 – 2/60 = -5/60
di = -60/5 = -12 cm
The image distance is negative, means that the image is virtual, where the beam of light does not pass through the image.
The image distance is 12 cm, smaller than the object distance of 30 cm.
画像の倍率（M）
M = -di / do = -(-12)/30 = 12/30 = 0.4
The magnification of image smaller than 1, means that the image size is smaller than the object size.
The height of image (h’)
画像の拡大率の式：
M = h'/h
h' = M h = (0.4)(10 cm) = 4 cm
The height of image is sign positive, means that the image is upright.
b) The focal length -20 cm and if the object distance (do) = 40 cm
画像距離（di）
1/di = 1/f – 1/do = -1/20 – 1/40 = -2/40 – 1/40 = -3/40
di = -40/3 = -13 cm
The image distance is negative, means that the image is virtual, where the beam of light does not pass through the image.
The image distance is 13 cm, smaller than the object distance 40 cm.
画像の倍率（M）
M = -di / do = -(-13)/40 = 13/40 = 0.3
The magnification of image smaller than 1, means that the image size is smaller than the object size.
The height of the image (h’)
The equation of the magnification of the image:
M = h'/h
h' = M h = (0.3)(10 cm) = 3 cm
The height of the image is signed positive, means that the image is upright.
