Convention of sign in Optics

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When an object is placed in front of a mirror or a lens, an image may be formed (when object is placed at the focal point of convex lens or concave mirror no image is formed). Now whether the image will be real or virtual and its position depend upon distance of the object, type of the lens or mirror and also their geometric construct. Yet it is required to establish a sign convention in order to find a relation between object distance, image distance and focal distance. There are a few sign conventions in Optcis, such as: Cartesian convention, real positive convention etc. Here we will discuss the real positive convention. Acoording to this one:

(i) Every distance has to be measured from the pole of reflecting or refracting plane or the optical center of the lens.

(ii) All real distances are positive. Here ‘real’ means the distance ray of light has actually traveled.

(iii) All virtual distances are negative. Here ‘virtual’ means the distance ray of light hasn’t actually traveled, but it seems to have traveled.

According to the above mentioned convention, following conclusions can be drawn:

(i) In case of the image formed by converging of rays of light after reflection or refraction, AKA real image, the concerned distance is positive. For similar reasons, distance of virtual image is negative. Because in order to form this, rays of light didn’t converge after reflection or refraction, they diverged. So frankly, no image is formed in this case, but the way reflected or refracted rays enter our eyes, we perceive to see an image that looks fully or partially like the object. That’s why virtual image can’t be focused on a screen, real image can.

(ii) In case of convex lens and concave mirrors, rays of light parallel to the principal axis are converged at a point on the axis, that’s why the focal lengths of these two are positive. Otherwise happens for concave lens and convex mirror, that’s why the focal points as well as the focal lengths of these two are negative. For similar reasons, when refraction takes place from lighter to denser medium the radii of curvature of concave and convex planes are respectively negative and positive. On the other hand, when refraction takes place from denser to lighter medium the radii of curvature of concave and convex planes are respectively positive and negative.

(iii) If object is situated in the same side from where the rays of light have arrived at the mirror, lens or refractive plane, the distance (u) of such object is positive. It the object is situated on the other side, the distance of such object is considered negative and the object itself is considered virtual or unreal. For example, consider refraction in the following coaxial convex lenses.

The principal axis of both the lens is the same.

An extended object is standing on the axis to the left of lens A. The image formed by this one will be acting as an object for lens B. However, it is to be noted that the object for lens B is NOT in the same side from where the rays of lights are coming. Hence this object for lens B is to be considered ‘virtual’ and its distance negative.

(iv) If image is situated in the same side where the rays of light should be after the reflection or refraction, the distance (v) of such image is positive. Otherwise, the distance of such image is negative and the image itself is called virtual or unreal. For example, consider the case presented above.

After the refraction has taken place, the rays of light has passed through the lens and they are now on the other side. For lens A, the image is formed in the same side where rays of light are after the refraction, that’s why this image is real.

If the ‘object’ for lens B lies within its focal length, it is going to form a virtual image which lies in the same side of the lens as the object, a case which should not happen, since light rays pass through the lens during refraction. So an image thus formed is to be called virtual or ‘unreal’.

Now consider reflection in mirrors. Since rays of light can’t pass through a mirror, rays of light are the front side of a mirror both before and after a reflection. In case of concave mirror, image may be formed in the front side, if the object is outside of focal point (u > f). In such a case the image is real and its distance (v) is positive. But image can also be formed in the back side of any type of mirror, where rays of light don’t go after the reflection. Such an image is virtual and its distance is considered negative.

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