### Question 22.
#### (a) Determining Object Distance, Focal Length, and Power of the Lens
To determine where the object should be placed, and the characteristics of the lens, follow these steps:
1. Identifying Given Data:
- The image distance (v) is given as 100 cm.
- The image is real and inverted, and the size of the image is twice the size of the object. This implies that the magnification (m) is -2 (since a real and inverted image produced by a lens has a negative magnification).
2. Finding Object Distance (u):
- Magnification (m) is given by the formula:
[tex]\[
m = -\frac{v}{u}
\][/tex]
- Substituting the values:
[tex]\[
-2 = -\frac{100}{u}
\][/tex]
[tex]\[
u = \frac{100}{2} = 50 \text{ cm}
\][/tex]
- Therefore, the object should be placed 50 cm in front of the lens.
3. Finding the Focal Length (f):
- The lens formula is:
[tex]\[
\frac{1}{f} = \frac{1}{v} - \frac{1}{u}
\][/tex]
- Substituting the known values:
[tex]\[
\frac{1}{f} = \frac{1}{100} - \frac{1}{50}
\][/tex]
[tex]\[
\frac{1}{f} = \frac{1}{100} - \frac{2}{100} = -\frac{1}{100}
\][/tex]
[tex]\[
f = -100 \text{ cm}
\][/tex]
4. Calculating the Power of the Lens (P):
- Power of the lens is given by:
[tex]\[
P = \frac{1}{f} \text{ (in meters)}
\][/tex]
- Converting focal length from cm to meters:
[tex]\[
f = -100 \text{ cm} = -1 \text{ m}
\][/tex]
- Therefore:
[tex]\[
P = \frac{1}{-1} = -1 \text{ diopters}
\][/tex]
Summary:
- Object distance (u) = 50 cm
- Focal length (f) = -100 cm
- Power of the lens (P) = -1 diopters
#### (b) Laws of Refraction
The laws of refraction, also known as Snell's laws, state:
1. First Law of Refraction:
- The incident ray, the refracted ray, and the normal to the interface of two media at the point of incidence all lie in the same plane.
2. Second Law of Refraction (Snell's Law):
- The ratio of the sine of the angle of incidence ([tex]\(\sin i\)[/tex]) to the sine of the angle of refraction ([tex]\(\sin r\)[/tex]) is a constant for the light of a given wavelength and for the given pair of media.
[tex]\[
\frac{\sin i}{\sin r} = \text{constant} = n
\][/tex]
- Where [tex]\(n\)[/tex] is the refractive index of the second medium with respect to the first medium.
