A magnifying glass uses a converging lens with a focal length of 14.5 cm. It produces a virtual and upright image that is 3.3 times larger than the object.
How far is the object from the lens?
To find the distance of the object from the lens, we can use the magnification formula:
Magnification (M) = -image distance (di) / object distance (do)
Given that the magnification (M) is 3.3 times larger than the object, we have:
M = 3.3 = -di / do
Since the image produced by the magnifying glass is virtual and upright, the image distance (di) will be positive. Therefore, we can rewrite the equation as:
3.3 = di / -do
Now, let's assume the image distance (di) is inf due to the virtual image produced by the magnifying glass. Substituting the values into the magnification formula:
3.3 = inf / -do
To solve for the object distance (do), we'll rearrange the equation:
do = inf / -3.3
Since dividing by infinity is undefined, we can conclude that the object distance (do) from the lens is also infinity, or the object is placed at a very far distance from the lens.
To find the distance of the object from the lens, we can use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens
- v is the image distance
- u is the object distance
Given:
- f = 14.5 cm (positive for converging lens)
- Image is virtual and upright, which means v is negative
- The image is 3.3 times larger than the object, which means the magnification (m) is 3.3
We can rearrange the lens formula to solve for the object distance (u):
1/f = 1/v - 1/u
Substituting the given values:
1/14.5 = 1/-v - 1/u
Since the image is 3.3 times larger, the magnification (m) is given by:
m = -v/u
3.3 = -v/u
We can now substitute this value and the given focal length (f) into the equation:
1/14.5 = 1/-v - 1/(3.3v)
Now we can solve for v by simplifying the equation and finding the common denominator:
1/14.5 = (3.3v - v)/(-3.3v)
Multiplying both sides by -3.3v to eliminate the denominators:
-3.3v/14.5 = 3.3v - v
Simplifying further:
-0.2269v = 2.3v - v
Combining like terms:
-0.2269v = 1.3v
Dividing both sides by 1.3v:
-0.2269 = 1.3
Solving for v:
v = -0.2269 / 1.3 ≈ -0.175 cm
Since v is negative, it indicates a virtual image. Now we can solve for u using the magnification formula:
m = -v/u
3.3 = -(-0.175)/u
Solving for u:
u = -(-0.175)/3.3 ≈ 0.053 cm or 0.53 mm
Therefore, the object is approximately 0.053 cm or 0.53 mm away from the lens.