A convex mirror has a focal length of -13 cm. A lightbulb with a diameter of 5.6 cm is placed 60.0 cm in front of the mirror.
a. Locate the image of the lightbulb.
b. Whats the diamter?
a. The image formed by a convex mirror is always virtual and upright. Since the focal length is negative (-13 cm), the image will be located behind the mirror. To find the location of the image, we can use the mirror equation:
1/f = 1/di + 1/do
Where f is the focal length, di is the image distance, and do is the object distance.
In this case, f = -13 cm and do = -60 cm (since the object is placed in front of the mirror). Plugging these values into the equation, we get:
1/-13 = 1/di - 1/-60
Simplifying, we find:
-60/-13 = 1/di + 1/60
4.615 = 1/di + 1/60
To solve for di, we rearrange the equation:
1/di = 4.615 - 1/60
1/di = (277 - 1)/60
1/di = 276/60
di = 60/276
di ≈ 0.217 cm
Therefore, the image of the lightbulb is located approximately 0.217 cm behind the convex mirror.
b. Since the image is virtual and upright, the diameter of the image will be the same as the diameter of the object. Therefore, the diameter of the image is 5.6 cm.
To locate the image of the lightbulb and determine its diameter, we can use the mirror formula and magnification equation.
a) First, let's use the mirror formula:
1/f = 1/di + 1/do
where:
f = focal length of the convex mirror (given as -13 cm)
di = image distance from the mirror (unknown)
do = object distance from the mirror, which is given as 60.0 cm
Substituting the values into the formula, we have:
1/-13 = 1/di + 1/60.0
Now let's solve for di:
1/di = 1/-13 - 1/60.0
1/di = -60/780 + 13/780
1/di = -47/780
di = -780/47
So, the image distance (di) is approximately -16.60 cm. Since the image distance is negative, it indicates that the image is virtual and located behind the mirror.
b) To find the diameter of the image, we can use the magnification equation:
magnification (m) = -di/do
where:
m = magnification
di = image distance (approximately -16.60 cm)
do = object distance (60.0 cm)
Substituting the values, we have:
m = -(-16.60)/60.0
m = 0.2767
The magnification of the image is positive, indicating that the image is upright and virtual.
Now, we can find the diameter of the image:
diameter of the image = diameter of the object x magnification
diameter of the image = 5.6 cm x 0.2767
diameter of the image ≈ 1.55 cm
Therefore, the diameter of the image of the lightbulb is approximately 1.55 cm.
To locate the image of the lightbulb formed by the convex mirror and determine its diameter, we can use the mirror equation and magnification equation.
a. Locating the image:
1. Begin by using the mirror equation:
1/f = 1/do + 1/di,
where f is the focal length, do is the object distance, and di is the image distance.
2. We know that the focal length (f) is -13 cm and the object distance (do) is 60.0 cm.
Substituting these values into the mirror equation, we get:
1/-13 = 1/60 + 1/di.
Simplifying this equation gives us:
-1/13 = 1/60 + 1/di.
3. Solve for the image distance (di) by rearranging the equation:
1/di = -1/13 - 1/60.
Combining the fractions on the right-hand side gives us:
1/di = -73/780.
To isolate di, take the reciprocal of both sides:
di = -780/73.
Thus, the image distance is approximately -10.68 cm.
4. Since the image distance (di) is negative, we conclude that the image formed by the convex mirror is virtual and located at a distance of approximately 10.68 cm behind the mirror surface.
b. Determining the diameter of the image:
To find the diameter of the image, we need to calculate the magnification (M) of the mirror.
1. The magnification is given by the equation:
M = -di/do,
where di is the image distance and do is the object distance.
2. Plug in the values:
M = -(-10.68 cm)/60.0 cm.
Simplifying the equation gives us:
M = 0.178.
3. The magnification is positive, indicating that the image is upright and smaller than the object.
4. To find the diameter of the image, multiply the magnification by the diameter of the object:
Diameter of the image = M * Diameter of the object.
Diameter of the image = 0.178 * 5.6 cm.
Diameter of the image is approximately 1 cm.
Therefore, the image of the lightbulb formed by the convex mirror is located approximately 10.68 cm behind the mirror surface, and its diameter is around 1 cm.
Use the magnification equation for the image diameter.
Use the lens equation for location.