A concave spherical mirror has a radius of
curvature of 29.4 cm . The object distance is
5.7 cm .Scale: 10 cm =
Find the magnitude of the image distance.
Answer in units of cm.
2- Find the magnification.
To find the magnitude of the image distance, we can use the mirror equation:
1/f = 1/do + 1/di
Where:
f is the focal length of the concave mirror, which can be calculated using the formula f = R/2, where R is the radius of curvature.
do is the object distance.
di is the image distance.
First, let's find the focal length (f) of the concave mirror:
f = R/2 = 29.4 cm / 2 = 14.7 cm
Now, we can substitute the values into the mirror equation to solve for the image distance (di):
1/14.7 = 1/5.7 + 1/di
To simplify the equation, we can find a common denominator:
1/14.7 = (5.7 + 1/di) / (5.7 * di)
Next, we can cross-multiply to remove the fractions:
14.7 * (5.7 + 1/di) = 5.7 * di
82.07 + 14.7/di = 5.7*di
Rearranging the equation, we get:
14.7/di = 5.7 * di - 82.07
Multiplying both sides by di:
14.7 = 5.7 * di^2 - 82.07 * di
Now, we have a quadratic equation. Rearrange to bring all terms to one side:
5.7 * di^2 - 82.07 * di + 14.7 = 0
To solve this quadratic equation, we can use the quadratic formula:
di = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 5.7, b = -82.07, and c = 14.7. Substituting these values into the quadratic formula:
di = (-(-82.07) ± √((-82.07)^2 - 4 * 5.7 * 14.7)) / (2 * 5.7)
Simplifying:
di = (82.07 ± √(6732.7249 - 416.52)) / 11.4
di = (82.07 ± √(6316.2049)) / 11.4
di = (82.07 ± 79.516) / 11.4
Using the positive root:
di = (82.07 + 79.516) / 11.4
di = 161.586 / 11.4
di ≈ 14.16 cm
Therefore, the magnitude of the image distance is approximately 14.16 cm.
2. To find the magnification (M), we can use the magnification formula:
M = -di/do
Where:
di is the image distance.
do is the object distance.
Substituting the given values:
M = -(14.16 cm) / (5.7 cm)
M ≈ -2.483
Therefore, the magnification is approximately -2.483.