The principal focus of a concave mirror is the point on its principal axis where rays parallel to the principal axis meet after reflection from the mirror.
For a spherical mirror, f = R/2. Given R = 20 cm, f = 20/2 = 10 cm.
The focal length is 10 cm.
A concave mirror can give an erect and enlarged image when the object is placed between the pole and the principal focus.
A convex mirror always forms a virtual, erect and diminished image, and it has a wider field of view than a plane mirror. Therefore a driver can see a larger area behind the vehicle.
For a spherical mirror, f = R/2 = 32/2 = 16 cm. For a convex mirror this focal length is positive by the Cartesian sign convention.
The focal length is 16 cm.
For a real image formed by a concave mirror, magnification is negative. Given m = -3 and u = -10 cm. Since m = -v/u, -3 = -v/(-10) = v/10, so v = -30 cm. Negative v means the image is in front of the mirror.
The image is located 30 cm in front of the mirror.
The ray bends towards the normal. Water is optically denser than air, so light slows down when it enters water from air and refracts towards the normal.
Refractive index n = c/v. Therefore v = c/n = (3 × 10⁸)/(1.50) = 2.0 × 10⁸ m s⁻¹.
The speed of light in glass is 2.0 × 10⁸ m s⁻¹.
From Table 9.3, diamond has the highest optical density because it has the highest refractive index, 2.42. Air has the lowest optical density because its refractive index is about 1.0003, the lowest in the table.
Light travels fastest in water. Water has the lowest refractive index among water, kerosene and turpentine, and lower refractive index means higher speed of light in that medium.
It means that light travels 2.42 times faster in vacuum/air than in diamond. Equivalently, speed of light in diamond = (speed of light in vacuum)/2.42.
One dioptre is the power of a lens whose focal length is 1 metre. Since P = 1/f (in metres), 1 D = 1 m⁻¹.
For a convex lens, a real inverted image equal to the object is formed when the object is at 2F and the image is at 2F on the other side. Given image distance = 50 cm = 2f, so f = 25 cm = 0.25 m. Power P = 1/f = 1/0.25 = +4 D.
The needle is placed 50 cm in front of the lens. The focal length is 25 cm and the power is +4 D.
For a concave lens, f = -2 m. Power P = 1/f = 1/(-2) = -0.5 D.
The power is -0.5 D.
- a. Water
- b. Glass
- c. Plastic
- d. Clay
A lens must be transparent. Clay is opaque, so it cannot be used to make a lens.
(d) Clay
- a. Between the principal focus and the centre of curvature
- b. At the centre of curvature
- c. Beyond the centre of curvature
- d. Between the pole of the mirror and its principal focus.
A concave mirror forms a virtual, erect and enlarged image only when the object is between P and F.
(d) Between the pole of the mirror and its principal focus.
- a. At the principal focus of the lens
- b. At twice the focal length
- c. At infinity
- d. Between the optical centre of the lens and its principal focus.
A convex lens forms a real image of the same size when the object is at 2F.
(b) At twice the focal length
- a. both concave.
- b. both convex.
- c. the mirror is concave and the lens is convex.
- d. the mirror is convex, but the lens is concave.
A concave mirror has negative focal length, and a concave lens also has negative focal length under the Cartesian sign convention.
(a) both concave.
- a. only plane.
- b. only concave.
- c. only convex.
- d. either plane or convex.
Plane and convex mirrors always form erect images; a concave mirror can form inverted images for many object positions.
(d) either plane or convex.
- a. A convex lens of focal length 50 cm.
- b. A concave lens of focal length 50 cm.
- c. A convex lens of focal length 5 cm.
- d. A concave lens of focal length 5 cm.
A magnifying glass is a convex lens. A smaller focal length gives higher power and greater magnification.
(c) A convex lens of focal length 5 cm.
The object should be placed between the pole and the principal focus, so its distance from the mirror should be less than 15 cm. The image formed is virtual, erect and larger than the object, and it is formed behind the mirror. In the ray diagram, one ray parallel to the principal axis reflects as if it comes from F, and another ray directed towards the centre of curvature reflects back along the same path; their backward extensions meet behind the mirror.
(a) Headlights of a car: concave mirror, because it can produce a strong parallel beam when the bulb is placed at its focus.
(b) Side/rear-view mirror: convex mirror, because it gives an erect diminished image with a wide field of view.
(c) Solar furnace: concave mirror, because it concentrates parallel sunlight at its focus to produce high temperature.
Yes, it will produce a complete image, but the image will be less bright. Each part of a convex lens receives light from every point of the object and can form the whole image. Covering half the lens reduces the amount of light passing through it.
For a convex lens, f = +10 cm, u = -25 cm, h = +5 cm. Lens formula: 1/v - 1/u = 1/f. So 1/v + 1/25 = 1/10, hence 1/v = 1/10 - 1/25 = 3/50 and v = 50/3 = 16.7 cm. Magnification m = v/u = 16.7/(-25) = -0.667. Image height h′ = mh = -0.667 × 5 = -3.3 cm.
The image is formed 16.7 cm on the other side of the lens. Its size is about 3.3 cm, and it is real, inverted and diminished.
For a concave lens, f = -15 cm and the virtual image is on the same side, so v = -10 cm. Lens formula: 1/v - 1/u = 1/f. Therefore -1/10 - 1/u = -1/15, so -1/u = 1/30 and u = -30 cm.
The object is placed 30 cm in front of the lens.
For a convex mirror, f = +15 cm and u = -10 cm. Mirror formula: 1/v + 1/u = 1/f. So 1/v - 1/10 = 1/15, hence 1/v = 1/15 + 1/10 = 1/6 and v = +6 cm. Positive v means the image is behind the mirror.
The image is formed 6 cm behind the mirror. It is virtual, erect and diminished.
The positive sign means the image is virtual and erect. The value 1 means the image is the same size as the object. In a plane mirror, the image is also formed as far behind the mirror as the object is in front.
For a convex mirror, R = 30 cm, so f = +15 cm. u = -20 cm, h = +5.0 cm. Mirror formula: 1/v + 1/u = 1/f, so 1/v - 1/20 = 1/15. Hence 1/v = 1/15 + 1/20 = 7/60 and v = 60/7 = 8.6 cm. Magnification m = -v/u = -8.6/(-20) = 0.43. Image height h′ = 0.43 × 5.0 = 2.1 cm.
The image is formed about 8.6 cm behind the mirror. It is virtual, erect and diminished, with size about 2.1 cm.
For a concave mirror, f = -18 cm, u = -27 cm, h = +7.0 cm. Mirror formula: 1/v + 1/u = 1/f. Thus 1/v - 1/27 = -1/18, so 1/v = -1/18 + 1/27 = -1/54 and v = -54 cm. Magnification m = -v/u = -(-54)/(-27) = -2. Therefore h′ = -2 × 7.0 = -14 cm.
The screen should be placed 54 cm in front of the mirror. The image size is 14 cm, and the image is real, inverted and enlarged.
Power P = 1/f, with f in metres. f = 1/P = 1/(-2.0) = -0.50 m. Negative focal length indicates a concave/diverging lens.
The focal length is -0.50 m, or -50 cm. It is a concave lens.
f = 1/P = 1/(+1.5) = +0.667 m. Positive power means a converging convex lens.
The focal length is +0.67 m, or about +66.7 cm. The lens is converging, so it is a convex lens.