WS Q2. Background A so-called thin lens is a lens taken in the approximation of it tending to zero thickness. Although we often illustrate it as a lens with thickness, this is just to relay its function. Strictly, we should draw a lens it as a plane under this approximation and you may choose to do this. We refer to this as the lens plane. It helps to be aware of this for the following question. A ray (see Th1.2) is defined by a position and a direction. In the following, you will consider how a ray just after a thin lens relates to a ray just before the lens. Q2. Rays (a) For a thin lens, write a straight forward equation for how the lateral position of a ray at plane 2 (just after the lens), x2, relates to the lateral position of a ray at plane 1 (just before the lens), x1. Hint: the answer is trivial. [2] (b) For a thin lens, determine (showing your steps) and simplify an equation for the angle of the ray just after the lens (plane 2), az in terms of the following three parameters, in the limit of small angles: ⚫ the angle of the ray just before the lens (plane1), α1; 1 2 ⚫ the lateral position of the ray just before the lens (plane 1), x1; ⚫ the focal length of the lens, f. [2] Guidance: first write down the thin lens equation in terms of u, v and f (as from Q1). Then substitute u and v in terms of a1, a2 and x1 using the following diagram and trigonometry simplified in the limit of small angles. In the limit of small angles, sin(a) -> a and tan(a) -> a, for a in radians. n = 1 f x2 Virtual x1 image Object α1 น V

WS Q2. Background A so-called thin lens is a lens taken in the approximation of it tending to zero thickness. Although we often illustrate it as a lens with thickness, this is just to relay its function. Strictly, we should draw a lens it as a plane under this approximation and you may choose to do this. We refer to this as the lens plane. It helps to be aware of this for the following question. A ray (see Th1.2) is defined by a position and a direction. In the following, you will consider how a ray just after a thin lens relates to a ray just before the lens. Q2. Rays (a) For a thin lens, write a straight forward equation for how the lateral position of a ray at plane 2 (just after the lens), x2, relates to the lateral position of a ray at plane 1 (just before the lens), x1. Hint: the answer is trivial. [2] (b) For a thin lens, determine (showing your steps) and simplify an equation for the angle of the ray just after the lens (plane 2), az in terms of the following three parameters, in the limit of small angles: ⚫ the angle of the ray just before the lens (plane1), α1; 1 2 ⚫ the lateral position of the ray just before the lens (plane 1), x1; ⚫ the focal length of the lens, f. [2] Guidance: first write down the thin lens equation in terms of u, v and f (as from Q1). Then substitute u and v in terms of a1, a2 and x1 using the following diagram and trigonometry simplified in the limit of small angles. In the limit of small angles, sin(a) -> a and tan(a) -> a, for a in radians. n = 1 f x2 Virtual x1 image Object α1 น V