A convex mirror is needed to produce an image that is nine-tenths the size of an object and located 21 cm behind the mirror. What focal length should be specified?
To determine the focal length of a convex mirror, we can use the mirror formula:
1/f = 1/v - 1/u
where:
- f is the focal length of the mirror,
- v is the image distance (distance of the image from the mirror),
- u is the object distance (distance of the object from the mirror).
In this case, we are given that the image is located 21 cm behind the mirror. Since the image is formed at the same side as the object in a convex mirror, the image distance (v) is negative (-21 cm).
We are also given that the image size is nine-tenths the size of the object. In a convex mirror, the image formed is always virtual, upright, and reduced in size. Since the image is smaller than the object, the magnification (m) is less than 1:
m = -v/u
Given that m = 9/10, we can substitute the values into the magnification equation:
9/10 = -(-21 cm) / u
(u is positive since the object is in front of the mirror)
Simplifying the equation:
9/10 = 21 cm / u
To solve for u, we can cross-multiply:
9u = 10 * 21 cm
9u = 210 cm
Dividing both sides by 9:
u = 210 cm / 9
u ≈ 23.33 cm
Now that we have the object distance (u) and the image distance (v), we can substitute these values into the mirror formula to find the focal length (f):
1/f = 1/v - 1/u
1/f = 1/(-21 cm) - 1/(23.33 cm)
Simplifying the equation:
1/f = -1/21 cm - 1/(23.33 cm)
1/f = -1/21 cm + (-1/23.33 cm)
Adding the fractions:
1/f ≈ -0.0476 - 0.0429
1/f ≈ -0.0905
To isolate f, we can take the reciprocal of both sides:
f ≈ 1/(-0.0905)
f ≈ -11.04 cm
However, the focal length of a convex mirror is always positive, so we take the absolute value of our result:
f ≈ | -11.04 cm |
f ≈ 11.04 cm
Therefore, the focal length that should be specified for the convex mirror is approximately 11.04 cm.
First of all the lens-maker's equation is correctly written as:
1/f = 1/u + 1/v,
And the appropriate sign convention is worked out for each specific case in question.
cases:
convex or concave lens, image distance is negative only for virtual images
(projected images) that is images formed on the same side of the lens.
plane or spherical mirror, image distance: v is negative, since image is virtual
Now, 1/u + 1/v => combine fractions by using the common denominator: uv
1/u + 1/v = (v + u)/uv = 1/f
f = uv/(u+v)
Note: always write the general lens equation with all positive terms,
the appropriate sign will come out in the wash.