Show mathematically that a diverging lens will alway produce an upright image. [i.e. prove that hi is positive if f is negative]
I'm not even sure how to start this. Can someone please help me?
Certainly! To mathematically show that a diverging lens will always produce an upright image, we can use the lens formula. The lens formula relates the object distance (u), the image distance (v), and the focal length (f) of the lens.
The lens formula is:
1/f = 1/v - 1/u
For a diverging lens, the focal length (f) is always negative. By convention, we take any distance to the left of the lens as negative (object distance, u) and any distance to the right of the lens as positive (image distance, v).
Now, let's assume that the object distance (u) is positive. In this case, to get a positive image distance (v), the second term on the right-hand side of the lens formula (1/u) must be greater than the first term (1/v).
Since the focal length (f) is negative, the term 1/f is also negative. This means that in order for 1/f to be greater than 1/v and make the lens formula work, the term 1/u must be negative.
For 1/u to be negative (because u is positive), we can conclude that the image distance (v) must also be negative. Therefore, the image produced by a diverging lens is always formed on the same side as the object, and it is an upright image because the image is formed in the opposite direction relative to the object.
So mathematically, if f is negative in the lens formula, the image distance (v) will be negative, indicating that the image formed by a diverging lens is always upright.