a) To determine whether the lens is converging or diverging, we observe the behavior of light passing through the lens. If the lens causes the light rays to converge towards a focal point, it is a converging lens. On the other hand, if the lens causes the light rays to diverge, it is a diverging lens. In this case, since an image is formed behind the lens, the lens is a converging lens.
b) To determine the focal length of the lens, we can use the lens formula:
1/f = 1/v - 1/u
where f is the focal length, v is the distance of the image from the lens, and u is the distance of the object from the lens.
Given that the object is located 30mm in front of the lens (u = -30mm) and the image is located 90mm behind the lens (v = -90mm), we can substitute these values into the formula:
1/f = 1/-90 - 1/-30
Simplifying:
1/f = -1/90 + 1/30
1/f = -2/90
1/f = -1/45
Taking the reciprocal of both sides:
f = -45mm
Therefore, the focal length of the lens is -45mm.
c) To draw a diagram with the lens at x = 0 and locate the image, we start by drawing a horizontal line representing the principal axis. Then, we draw a vertical line perpendicular to the principal axis at the position of the lens.
On the left side of the lens, we mark the object distance of 30mm (-30mm from the lens) from the lens. On the right side of the lens, we mark the image distance of 90mm (-90mm from the lens). We can represent the lens as a simple vertical line bisected by the principal axis.
d) The characteristics of the image can be determined by analyzing its properties:
- Real or virtual: Since the image is formed behind the lens, it is a real image.
- Size (larger, smaller, or the same): Since the image distance (-90mm) is greater than the object distance (-30mm), the image is larger than the object.
- Orientation (inverted or upright): Since the image is formed on the opposite side of the lens compared to the object, it is inverted in comparison to the object.
e) To draw the concave mirror at x = 0, we start by drawing a horizontal line representing the principal axis. Then, we draw a vertical line perpendicular to the principal axis at the position of the mirror.
Since the concave mirror forms a real image, we need to draw at least two rays:
- Incident ray parallel to the principal axis: We draw a ray coming from the object parallel to the principal axis, which reflects off the mirror and passes through the focal point.
- Incident ray passing through the focal point: We draw a ray coming from the object towards the focal point, which reflects off the mirror and becomes parallel to the principal axis.
The intersection point of these two rays will determine the location of the image formed by the concave mirror.
f) To calculate the height of the image and the magnification of the mirror, we can use the mirror formula:
1/f = 1/v - 1/u
where f is the focal length, v is the image distance, and u is the object distance.
Given that the focal length of the concave mirror is 20mm and the object distance is 30mm, we can substitute these values into the formula:
1/20 = 1/v - 1/30
Simplifying:
1/v = 1/20 + 1/30
1/v = 3/60 + 2/60
1/v = 5/60
1/v = 1/12
Taking the reciprocal of both sides:
v = 12mm
Now, to calculate the height of the image, we can use the magnification equation:
m = -v/u
where m is the magnification, v is the image distance, and u is the object distance.
Substituting the given values:
m = -12/30
m = -0.4
Therefore, the height of the image is -0.4 times the height of the object, indicating that the image is smaller in size.