A converging lens produces a four times magnified and upright image of an object placed in front of it. If the focal length of the lens is 20cm. Calculate the object distance.
To calculate the object distance in this scenario, we can use the magnification formula:
Magnification (m) = -image distance (di) / object distance (do)
Given that the magnification (m) is 4, we can substitute it into the formula:
4 = -di / do
Since the image is upright, the negative sign is used to indicate that the image is virtual.
Now, let's determine the focal length (f) of the converging lens, which is given as 20 cm. For a converging lens, the focal length is positive.
Using the lens formula:
1/f = 1/do + 1/di
Substituting the given values:
1/20 = 1/do + 1/di
We can simplify the equation to:
1/do = 1/20 - 1/di
Now, we have two equations:
4 = -di / do
1/do = 1/20 - 1/di
To solve these equations, we need to combine them and eliminate one of the variables.
Multiplying both sides of the first equation by di and multiplying both sides of the second equation by do, we obtain:
4 * di = -do
do = 20 - (20 / di)
Substituting do = 20 - (20 / di) into the first equation:
4 * di = -(20 - (20 / di))
Expanding the equation:
4 * di = -20 + (20 / di)
Multiplying both sides of the equation by di:
4 * di^2 = -20di + 20
Rearranging the equation:
4di^2 + 20di - 20 = 0
Now we have a quadratic equation. We can solve it by factoring or by using the quadratic formula.
In this case, the quadratic equation can be simplified by dividing all terms by 4:
di^2 + 5di - 5 = 0
To find the roots of the equation, we can use factoring, completing the square, or the quadratic formula.
Using the quadratic formula:
di = (-b ± √(b^2 - 4ac)) / (2a)
With a = 1, b = 5, and c = -5, we can substitute these values into the quadratic formula:
di = (-5 ± √(5^2 - 4(1)(-5))) / (2(1))
Simplifying further:
di = (-5 ± √(25 + 20)) / 2
di = (-5 ± √45) / 2
So, we have two possible values for the image distance (di):
di = (-5 + √45) / 2
di = (-5 - √45) / 2
Since the image is virtual and upright, the distance must be positive. Thus, only the positive solution is valid:
di = (-5 + √45) / 2
Now we can substitute this value back into the equation to find the object distance (do):
do = 20 - (20 / di)
do = 20 - (20 / ((-5 + √45) / 2))
Simplifying further, we get the object distance:
do ≈ -3.05 cm
Therefore, the object distance is approximately -3.05 cm. However, since the object is placed in front of the lens, the object distance should be positive. Hence, in the actual scenario, the object distance is 3.05 cm.