5. An object is 30.0 cm to the left of a convex lens with a focal length of +8.0 cm. Draw a ray diagram of the setup showing the location of the image. Calculate the distance of the image from the lens. Is the image upright or inverted?
To draw a ray diagram and determine the distance of the image from the lens, we can use the lens formula:
1/f = 1/v - 1/u
Where:
- f = focal length of the lens
- v = distance of the image from the lens (positive if on the same side as the object, negative if on the opposite side)
- u = distance of the object from the lens (positive if on the same side as the object, negative if on the opposite side)
Given:
- f = +8.0 cm (convex lens)
- u = -30.0 cm (to the left of the lens)
Let's plug in the values and solve for v:
1/8.0 cm = 1/v - 1/-30.0 cm
Simplifying the equation, we get:
1/8.0 cm = 1/v + 1/30.0 cm
We can find the common denominator by multiplying the denominators:
1/8.0 cm = (30/30v) + (v/30v)
1/8.0 cm = (v + 30v)/(30v)
Now, we can cross-multiply:
30v = 8.0 cm * (v + 30v)
Expanding the equation:
30v = 8.0 cm * v + 240 cm
Subtracting 8.0 cm * v from both sides:
22v = 240 cm
Now, divide by 22 on both sides:
v ≈ 10.91 cm
From the calculation, we find that the distance of the image from the lens is approximately 10.91 cm. Since it is positive, the image is formed on the same side as the object.
To determine if the image is upright or inverted, we need to look at the position of the object and the image. Since the object is on the left side of the lens, and the image is also on the left side, the image is virtual and upright.
To draw a ray diagram for the given setup, follow these steps:
Step 1: Draw a vertical line representing the principal axis of the lens.
Step 2: Draw an arrowhead to represent the object on the left side of the lens, 30.0 cm away from it.
Step 3: Draw a ray from the top of the object parallel to the principal axis.
Step 4: Draw a ray from the top of the object passing through the focal point on the opposite side of the lens and refract it towards the principal axis.
Step 5: Draw a ray from the top of the object that passes through the center of the lens and continues undeviated.
Step 6: Repeat steps 3-5 for the bottom of the object.
Step 7: The point where these two refracted rays intersect is the location of the image.
Now, to calculate the distance of the image from the lens, you can use the lens equation:
1/f = 1/di - 1/do
Where:
f = focal length of the lens (+8.0 cm)
di = image distance (unknown)
do = object distance (-30.0 cm)
Plugging in the values and solving for di:
1/8.0 cm = 1/di - 1/-30.0 cm
Simplifying the equation:
1/8.0 cm = 1/di + 1/30.0 cm
Multiplying through by 8di:
di = (8 × 30) / (8 + 30)
di = 240 / 38
di ≈ 6.32 cm
Therefore, the distance of the image from the lens is approximately 6.32 cm.
To determine if the image is upright or inverted, we can use the magnification equation:
m = -di / do
Where m is the magnification.
Plugging in the values:
m = -6.32 cm / -30.0 cm
Simplifying the equation:
m ≈ 0.21
Since the magnification is positive, the image is upright.
Sorry, we don't do drawings, including ray diagrams, here. Your textbook should explain how it is done. If not there are plenty of web sites that do.
For the location of the image, Di, you can always use the equation
1/Do + 1/Di = 1/f,
with
Do = 30 and f = 8
The image should be real and inverted in this case