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?
You get an upright (virtual) image if the magnification is a negative number. The magnification is the ratio of the image distance si to the object distance so. The lens equation says that
1/so + 1/si = 1/f
If f is negative and so is positive, si must be negative. Therefore si/so must be negative also.
okay, but I'm not sure how that explains it algebraicall, could you please elaborate?
If 1/f is negative (as it is for a diverging lens), both so and si cannot be positive. That is the best is can do with the algebra.
You are restricted to positive values so (the object distance) with the source in front of the lens.
Sure! I'll guide you through the process of showing mathematically that a diverging lens will always produce an upright image.
To start, let's define the variables:
- hi: The height of the object.
- ho: The height of the upright image.
- f: The focal length of the diverging lens.
The lens formula relates these variables and can be written as:
1/hi + 1/ho = 1/f
Since we're interested in proving that hi is positive when f is negative (upright image), we need to substitute the given values into the lens formula and solve for hi. Let's assume ho is also positive, indicating an upright image.
Given that f is negative for a diverging lens, let's assign it a value, say -f. Now we can rewrite the lens formula as:
1/hi + 1/ho = -1/(-f)
Next, let's simplify the equation by finding the reciprocal of both sides:
1/hi = (-1/-f) - 1/ho
1/hi = 1/f - 1/ho
Now, let's further simplify by finding the common denominator:
1/hi = (ho - f)/(f * ho)
To isolate hi, we can take the reciprocal of both sides:
hi = (f * ho)/(ho - f)
Now, we can analyze the equation. Since ho is positive (upright image) and f is negative (diverging lens), the numerator (f * ho) will be negative.
For the denominator (ho - f), if we subtract a negative number (f) from a positive number (ho), we'll end up with a positive value as well.
So, we have a negative value (numerator) divided by a positive value (denominator), resulting in a positive value for hi.
Therefore, mathematically, we have proved that hi is positive when f is negative, showing that a diverging lens will always produce an upright image.