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How to find the focal length of convex mirror using convex lens?
We cannot use an optical method to determine the focal length of a convex mirror without using a convex (converging) lens since a convex mirror does not form a real image of an object in front of it or even parallel rays coming from infinity.
Therefore we have to use a convex mirror to determine the focal length of the convex mirror by an indirect method.
STEP 1 DETERMINE THE ROUGH FOCAL LENGTH OF THE CONVEX LENS
First of all we take a convex lens and determine its rough focal length by forming a real image of a distant object on a screen. The distance between the convex lens and the screen gives the rough focal length, since when the object is at infinity, the image formed by the convex lens is at its focus. This method is called the distant object method to estimate the focal length of a convex lens. The focal length of a concave mirror may also be estimated using this method.
Now, as we know the approximate focal length of the convex lens, keep an object- an optical pin or a lit candle in front of the convex lens at a distance around 2 times the rough focal length determined and place a screen on the other side so as to form a real image of the object on it. This adjustment is done so as to make the distances involved quite manageable.
When you get a clear and sharp image on the screen, mark the positions of the convex lens and the screen on the table.
Place the convex mirror in between the screen and the lens without disturbing the position of the candle and the lens.
Place the screen close to the candle. Carefully adjust the position of the convex mirror alone so that you get a sharp image on the screen which is now kept at the position of and along with the candle. Mark the position of the convex mirror now.
The distance between the position of the convex mirror and the old position of the screen gives the radius of curvature of the convex mirror. (Since r = 2f; we can determine the focal length by dividing the distance with 2)
Why the distance is r (2f)?
We know that when a ray of light is incident normally on the surface of a mirror, it retraces its path.
We also know that a line drawn from the centre of curvature of the mirror to the surface of a mirror is normal to the surface.
Therefore, when we are getting the image at the same position as the candle, it is formed by retracing of the rays, which means that the old position of the screen is at the position of the centre of curvature of the convex lens.
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See the Videos below to understand how the focal length of a convex mirror is determined using a convex lens in an optic bench.