General Physics II Optical Instruments 1
The Thin-Lens Equation 2
The Thin-Lens Equation Using geometry, one can show that 1 1 1 s+ =. s' f The magnification of the lens is defined by For a thin lens, one has M = h'. h M = ' s. (Thin-lens Equation) 3
The Thin-Lens Equation Sign Convention for Lenses (and Mirrors) Object Distance: For the situations to be dealt with in this course, the object distance is always positive. Image Distance: When the image is on the opposite side of the lens as the object, the image distance (s ) ispositive. In this case, the image is real. When the image is on the same side of the lens as the object, the image distance (s ) is negative. In this case, the image is virtual. Focal Length: The focal length (f) for a converging lens is positive. The focal length for a diverging lens is negative. Object and Image Heights: The height of the object (h) is positive. If the image is upright, then its height (h ) is positive and so the magnification (M) is positive. If the image is inverted, its height is negative and so the magnification is negative. 4
Image Formation by Two Thin Lenses Overall magnification M = M 1 M 2. 5
Refractive Power The refractive power of a lens measures it ability to bend light by refraction. The refractive power is defined as the reciprocal of its focal length of the lens: P = 1. f If f is in meters, then the refractive power is in diopters. A converging lens has a positive refractive power; a diverging lens has a negative power. Lenses with greater surface curvature and/or greater index of refraction will have a smaller focal length (absolute value) and hence a greater refractive power (absolute value). 6
Workbook: Chapter 19, Questions 1, 3 Textbook: Chapter 19, Problem 39. Group Problem Solving: Problem 42 7
The Camera A camera produces a real (inverted) image of an object. In a digital camera, a chargedcoupled device (CCD) detector is located at the position of the image in order to record it. (Older cameras use film.) Since the image is real, the overall action of all the lens elements must be that of a converging lens. The lens-detector distance is adjusted to focus on objects at varying distances from the camera. 8
The Human Eye The cornea, aqueous humor, and the lens refract the incoming light to produce a sharp real image on the retina. The eye can quickly focus on objects at vastly different distances by a process called accommodation. The eye performs this function by changing the shape of the lens. 9
The Human Eye The near point of an eye is the smallest distance at which the eye can focus on an object and see it clearly. The lens has its greatest curvature (smallest focal length). For a normal eye, the near point is taken by convention to be 25 cm from the eye. The far point of an eye is the greatest distance at which the eye can focus on an object and see it clearly. The eye is completely relaxed and the lens has its smallest curvature (greatest focal length). For a normal eye, the far point is at infinity. 10
Workbook: Chapter 19, Question 6 11
Nearsightedness Nearsightedness (or myopia) occurs when the eye cannot focus on distant objects. Nearly parallel rays from a distant object are brought to a focus in front of the retina because the lens has too much curvature when fully relaxed or the eye ball is too long. Thus, a sharp image is formed in front of the retina, not on the retina. The far point of a myopic eye is less than infinity. Objects closer than the far point can be seen clearly. 12
Farsightedness Farsightedness (or hyperopia) occurs when the eye cannot focus on objects that are nearby. Diverging rays from a nearby object have not yet converged to a focus when they reach the retina because the lens does not have enough curvature or the eye ball is too short. Thus, no sharp image is formed on the retina. The near point of a hyperopic eye is greater than 25 cm. Objects farther away than the near point can be seen clearly. 13
Correction of Nearsightedness To correct nearsightedness, lights rays from a distant object must be made to come to a focus on the retina. Thus, these rays need to converge less rapidly. This can be achieved with the aid of a diverging lens (glasses or contacts). For an object at infinity, the diverging lens forms a virtual image at the eye s far point. The relaxed eye now sees the distant object clearly by focusing on this virtual image. 14
Correction of Farsightedness To correct farsightedness, lights rays from a nearby object must be made to come to a focus on the retina. Thus, these rays need to converge more rapidly. This can be achieved with the aid of a converging lens (glasses or contacts). With the object at the normal near point, the converging lens produces a virtual image at the eye s actual near point. The eye can clearly see this image. 15
Textbook: Chapter 19, Problem 20 16
The Magnifier During a solar eclipse, the Moon totally covers the Sun. Is the Moon the same size as the Sun? Of course not. The Moon is a lot closer to Earth than the Sun. The Moon is also a lot smaller than the Sun. Because of the differences in distance and size, the Moon and the Sun subtend approximately the same angle to the viewer s eye. Thus, both seem to be the same size. 17
A converging lens can produce a magnified virtual image of an object by increasing the angle subtended by the image to the eye relative to that subtended by the object without the lens. If the object is placed very close to (and inside) the focal point of the lens, the angular magnification is given by M = θ = h/ f = 25 cm. θ h/25 cm f 0 The focal length f is in cm. The Magnifier θ tan θ = h. 0 0 25 cm ( angles must be small and in radians!) θ tan θ = h. f 18
The Microscope The microscope uses two lenses: (1) an objective, which creates an enlarged intermediate real image of a nearby object; and (2) an eyepiece, which is used as a magnifier to magnify the intermediate image. M = M M = o e L f o 25 cm. f e (Note: M o = -s /s -L/f o.) 19
The Refracting Telescope The telescope uses two lenses: (1) an objective, which creates an intermediate real image of a distant object; and (2) an eyepiece, which is used as a magnifier to magnify the intermediate image. M = θ = h / f = f θ h / f f e o o o e e. 20