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Mirror Equation visualized

We use mirrors all the time. We have seen that the image distance is same as the object distance in case of plain mirrors. How about image distance for Spherical mirrors ? The mirror equation helps. We can verify this relationship with the help of optical bench OR we can use ray paths for the same. We will use this acrylic mirror strip and bend it to get a concave profile. This card sheet is glued to a cardboard sheet. Circle is marked with a 25 centimeter radius. Gap is carved in the base and a mirror strip is locked in place. This is the principal axis. This is Pole point P . This is focal point F. This is the center of curvature C. Focal length is half the radius of curvature for concave mirror, that is 12.5 centimeter. If an object is placed between focal point and center of curvature or beyond center of curvature, we get a real image. Let us see if we can verify the mirror equation. We will use flashlight from these cell phones. Slit is formed with the help of blades. One ray will be projected parallel to the principal axis. Other ray will connect the tip of the pencil and pass through the focal point. Image tip will be formed where two reflected rays meet. Let us place this pencil on the center of curvature. We will project the first ray parallel to the principal axis. We will project the second ray which connects the tip of the pencil and passes through the focal point. Image tip is a point where two reflected rays meet. Let us place this another pencil here. This is our image pencil. Let us find out focal length from the formula. Image distance and object distance from the pole are the same. Cartesian sign conventions We will use new cartesian sign convention to verify. This is pole P. Principal axis is our x axis. All the distances to the left side of the pole are taken as negative.All distances to the right side of the pole are taken as positive. Distances above the principal axis or X axis are positive. Distances below the principal or X axis are negative. Mirror Formula Our object is 6 cm long and placed on the center of curvature. Lets call it u. Value of u is -25 cm as the object is on the left side from pole p. Image formed is exactly below this object at the same point. This distance is called as v. Its value is -25 cm. Focal length is marked as f. As per the formula. 1/u + 1/v = 1/ f U + V / UV = 1 / f F = UV / ( U + V) F = 25 * 25 / ( -50 ) = - 12.5 cm As the focal point is on the left side of the pole, it has negative value as per convention. We can do similar exercise when Pencil is placed between center of curvature and focal point as well as beyond center of curvature.