Prove that if an object is at a distance of twice the focal length from a lens the image is the same size as the object. Show this using an algabraic method.
To prove that if an object is at a distance of twice the focal length from a lens, the image is the same size as the object, we can use the lens equation and magnification equation.
The lens equation states:
1/f = 1/do + 1/di
where:
- f is the focal length of the lens,
- do is the object distance (distance of the object from the lens), and
- di is the image distance (distance of the image from the lens).
The magnification equation states:
m = -di/do
where m is the magnification.
Let's assume that the object height (or size) is denoted by ho and the image height is denoted by hi.
Since we want to prove that the image is the same size as the object, we need to show that hi = ho.
Now, let's solve this algebraically:
1. Start with the lens equation:
1/f = 1/do + 1/di
2. Substitute do = 2f (object distance is twice the focal length):
1/f = 1/(2f) + 1/di
3. Simplify:
1/f = (1 + 2)/(2f) [common denominator]
4. Simplify further:
1/f = 3/(2f)
5. Cross-multiply:
2f = 3f
6. Cancel out the f term by dividing both sides by f:
2 = 3
Since step 6 leads to an inconsistent equation (2 equals 3), our assumption that the image is the same size as the object when the object distance is twice the focal length is proven to be incorrect.
Therefore, we cannot prove using an algebraic method that if an object is at a distance of twice the focal length from a lens, the image is the same size as the object.