A convex mirror with a radius of curvature of 39.0 cm forms a 0.95 cm tall image of a pencil at a distance of 12.1 cm behind the mirror. What is the magnification?
To find the magnification of a convex mirror, we can use the mirror equation:
1/f = 1/o + 1/i
where:
f = focal length of the mirror
o = object distance from the mirror (positive in front of the mirror)
i = image distance from the mirror (positive behind the mirror)
In the case of a convex mirror, the focal length is negative. So, let's first find the focal length using the given radius of curvature (R):
R = 2 * f
Therefore, f = R / 2 = 39.0 cm / 2 = 19.5 cm
Now, we can use the mirror equation to find the image distance (i):
1/19.5 cm = 1/o + 1/12.1 cm
Let's solve for 1/o:
1/o = 1/19.5 cm - 1/12.1 cm
= (12.1 cm - 19.5 cm) / (19.5 cm * 12.1 cm)
= -7.4 cm / (19.5 cm * 12.1 cm)
= -7.4 cm / 236.45 cm^2
= -0.03126 cm^(-1)
To find the magnification (M), we can use the formula:
M = -i / o
Let's substitute the values:
M = -(-0.03126 cm^(-1)) / o
Now, we need to find the image distance (i) to calculate the magnification. We know the height of the image (h_i) and we can use the magnification formula:
M = h_i / h_o
where:
h_i = height of the image
h_o = height of the object
The height of the object (h_o) is given as 0.95 cm.
Therefore, we can rearrange the magnification formula to calculate the image distance (i):
i = M * o
Now, let's substitute the values to calculate i:
i = -(-0.03126 cm^(-1)) / (-0.03126 cm^(-1)) * 12.1 cm
i = 12.1 cm
Now that we have the image distance (i), we can calculate the magnification (M):
M = -i / o
M = -12.1 cm / 12.1 cm
M = -1
Hence, the magnification of the convex mirror is -1.