Find the size and the nature of the image. The origin of the Cartesian coordinate system is at the center of the optical component. Drawing the ray diagram: Using a scale of 1: 5, we get v = - 2 cm, f = - 3 cm. How far is the object placed from the lens? The pole (P) of the mirror is considered as the origin and the principal axis of the mirror is taken as the x-axis. According to the new Cartesian sign convention for the refraction of light through the concave and convex lens. (vi) Draw a line from A to C (centre of the lens), which goes straight without deviation. Fig: Cartesian Sign Convention Sign convention is a set of rules to set signs for image distance, object distance, focal length, etc for mathematical analysis of image formation. The sign is taken as−(negative) from the pole of a spherical mirror towards the object along the principle axis. The heights measured upwards and perpendiculars to the principal axis of the mirror are taken as positive. An object 5 cm in length is held 25 cm away from a converging lens of focal length 10 cm. The image is real, inverted and enlarged in size. (xi) Thus the final position, nature and size of the image A'B' are: (a) Position of image A'B' = 3.3 cm × 5 = 16.5 cm from the lens on opposite side. According to Cartesian sign convention, the focal length of a lens is given by. All the distances in a ray diagram of reflection by spherical mirrors are measured from the pole of the spherical mirror. (ix) The AB is position of object. According to it: Object is always placed to the left of mirror; All distances are measured from the pole of the mirror. An object of size 7.0 cm is placed at 27 cm in front of a concave mirror of focal length 18 cm. (b) Nature of image A’B’: Real and inverted. According to Cartesian sign convention, the focal length of a lens is given by. Sign is taken as + (positive) behind the spherical mirror. Here, Object size, h = + 7.0 cmObject distance, u = - 27 cmFocal length, f = - 18 cm Image distance, v = ? 232, Block C-3, Janakpuri, New Delhi, Sign convention - definition Sign convention is a set of rules to set signs for image distance, object distance, focal length, etc for mathematical analysis of image formation. (3) The distances measured in the direction opposite to the direction of incident light are taken as negative. (2) The distances measured in the same direction as the incident light, are taken as positive. Focal length, f = - 15 cm [f is - ve for a concave lens], Image distance, v = - 10 cm [Concave lens forms virtual image on same side as the object, so v is - ve]. (7) The magnification is positive for the erect (and virtual) image and negative for an inverted (and real) image. Therefore, on this scale 5 cm high object, object distance of 25 cm and focal length of 10 cm can be represented by 1 cm high, 5 cm and 2 cm lines respectively. The point of intersection of the incident rays is called object and the point of intersection of reflected light rays is called image. The height of is taken as + (positive) above the principal axis and taken as – (negative) below the principal axis. (v) Draw a line AD parallel to principal axis and then, allow it to pass straight through the focus (F') on the right side of the lens. (ii) Centre line is shown by DE. (viii) Draw a line AB, perpendicular (downwards) from A to meet the principal axis at B. (v) Draw a line AD, parallel to principal axis. As the distances given in the question are large, so we choose a scale of 1: 5, i.e., 1 cm represents 5 cm. Focal length, f = - 15 cm [f is - ve for a concave lens]Image distance, v = - 10 cm [Concave lens forms virtual image on same side as the object, so v is - ve]As, Object distance, u = -30 cm. (vii) Let the two lines starting from A meet at A'. (6) The size of the object is always taken as positive, but the image size is positive for erect image and negative for an inverted image. (c) Height of image A'B': 0.7 × 5 = 3.5 cm, i.e., image is smaller than the object. © It will be found to be equal to 6 cm.Thus, object is placed at a distance of 6 cm × 5 = 30 cm from the lens. If a virtual image is formed, image is formed on left side, so image distance is negative Since object is always above the principal axis, object height will be positive If image is above the principal axis, image height will be positive. All distances are measured from the optical centre of the mirror. Draw the ray diagram. Now, using the mirror formula, 1u+1v = 1f∴ 1v = 1f-1u ⇒ = 1-18-1-27 = -3+254 = -154i.e., v = -54 cm The screen should be placed at a distance of 54 cm on the object side of the mirror to obtain a sharp image. Find the position of the image, its nature and size. This means the sign is always taken as −(negative) in front of a spherical mirror. All rights reserved. A virtual and erect image of height 2.2 cm is formed behind the mirror (because v is positive) at a distance of 8.6 cm from the mirror. © copyright 2020 QS Study. This sign convention is known as New Cartesian Sign Convention. (viii) Draw AB', perpendicular to the principal axis from A'. Image size, h' = ? It is also known as a mirror formula. All distances measured to the right of the origin are taken as positive while those measured to the left of the origin are taken as negative. (ix) Now AB', represents the real, but inverted image of the object AB. The object is always placed to the left of the mirror. Magnification, m = -vu= h'h Therefore, Image size, h' = -vhu = -8.6 × 5-20 = 2.15 ≃ 2.2 cm. All distances parallel to the principal axis are measured from the pole of the mirror. (iv) Join any point D (nearly at the top of lens) and F by a dotted line. For example if an image is formed behind the mirror, the distance of image is taken as + (positive) from pole along the principal axis. A virtual and erect image of height 2.2 cm is formed behind the mirror (because v is positive) at a distance of 8.6 cm from the mirror. To measure object distance, image distance, focal length and radius of curvature we need to follow certain sign convention. (iii) Mark two foci F and F' on two sides of the lens, each at a distance of 2 cm from the lens. When a point object is placed before a spherical mirror, a point images formed. (4) Heights measured perpendicular to the principal axis, in the upward direction are taken as positive.

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