1. Physics 7E
Prof. D. Casper

2. Admin Chapter 33 HW is due Thursday, 7 am
Discussion Thursday (Chapter 34)
Reading
Next Monday: Chapter 34.5 – 34.8
Next Wednesday: Chapter 35.1 – 35.3
Week from Monday: Chapter 35.4 – 35.5
Week 8 9 10
Finals
Monday
Tuesday
Wednesday
Thursday
Friday
Today
34.5 – 34. 8
35.1 – 35.3
Thanksgiving
35.4 – 35.5 Ch. 34 HW Due
Ch. 36 Lecture
Ch. 35 HW Due
Ch. 36 HW Due
Final (8am)

3. Spherical Mirrors: Sign Convention
Concave Mirrors: Radius R is positive Convex Mirrors: Radius R is negative

4. Spherical Mirrors: Paraxial Approximation
Assume 𝛼,𝛽,𝜙 are small angles Then 𝛿≪𝑠, 𝑠 ′ ,𝑅 tan 𝛼 = ℎ 𝑠−𝛿 →𝛼≈ ℎ 𝑠 tan 𝛽 = ℎ 𝑠 ′ −𝛿 →𝛽≈ ℎ 𝑠 ′ tan 𝜙 = ℎ 𝑅−𝛿 →𝜙≈ ℎ 𝑅 𝛼+𝛽=2𝜙 1 𝑠 + 1 𝑠 ′ = 2 𝑅
“Exact” ray tracing for a concave mirror

5. Spherical Mirrors: Paraxial Approximation
“Exact” ray tracing for a concave mirror
Paraxial approximation

6. Spherical Mirrors: Lateral Magnification

7. Concave Mirror: Focal Point
1 𝑠 + 1 𝑠 ′ = 1 𝑓
Reminder: Rays are reversible

8. Concave Mirror: Ray Tracing

9. Convex Mirror: Focal Point
Here: R < 0 , f < 0 and s < 0
Note: R < 0 , f < 0 and s’ < 0

10. Convex Mirror: Ray Tracing
1 𝑠 + 1 𝑠 ′ = 1 𝑓 → 1 𝑠 ′ = 1 𝑓 − 1 𝑠
if 𝑓<0 and 𝑠>0, then 𝑠 ′ <0
Real object always forms virtual image in convex mirror

11. Concave Mirror: Real and Virtual Images
In these examples: 𝑅=20 cm, 𝑓=10 cm
𝑠=30 cm: 𝑠 ′ = 1 10 → 𝑠 ′ =15 cm 𝑚=− 𝑠 ′ 𝑠 =− 1 2 (real, inverted image)
𝑠=20 cm: 𝑠 ′ = 1 10 → 𝑠 ′ =20 cm 𝑚=− 𝑠 ′ 𝑠 =−1 (real, inverted image)
𝑠=10 cm: 𝑠 ′ = 1 10 → 𝑠 ′ →∞ (no image)
𝑠=5 cm: 𝑠 ′ = 1 10 → 𝑠 ′ =−10 cm 𝑚=− 𝑠 ′ 𝑠 =+2 (virtual, upright image)

12. A. real and larger than the object.
Q34.3
An object is placed 4.0 m away from a concave mirror of focal length +1.0 m. The image formed by the mirror is
A. real and larger than the object.
B. real and smaller than the object.
C. real and the same size as the object.
D. virtual and larger than the object.
E. virtual and smaller than the object.
Answer: B

13. A. real and larger than the object.
An object is placed 4.0 m away from a concave mirror of focal length +1.0 m. The image formed by the mirror is
A. real and larger than the object.
B. real and smaller than the object.
C. real and the same size as the object.
D. virtual and larger than the object.
E. virtual and smaller than the object.

14. A. real and larger than the object.
Q34.7
An object is placed 2.0 m away from a convex mirror of focal length –1.0 m. The image formed by the mirror is
A. real and larger than the object.
B. real and smaller than the object.
C. real and the same size as the object.
D. virtual and larger than the object.
E. virtual and smaller than the object.
Answer: E

15. A. real and larger than the object.
An object is placed 2.0 m away from a convex mirror of focal length –1.0 m. The image formed by the mirror is
A. real and larger than the object.
B. real and smaller than the object.
C. real and the same size as the object.
D. virtual and larger than the object.
E. virtual and smaller than the object.

16. Spherical Refracting Surface: Paraxial Approximation
𝑛 𝑎 𝑠 + 𝑛 𝑏 𝑠 ′ = 𝑛 𝑏 − 𝑛 𝑎 𝑅
𝑚= 𝑦 ′ 𝑦 =− 𝑛 𝑎 𝑠 ′ 𝑛 𝑏 𝑠
Center on side with outgoing light: 𝑅>0
Center not on side with outgoing light: 𝑅<0

17. Lens Properties Thicker in middle focal length 𝑓>0
Thinner in middle
focal length 𝑓<0

18. Thin Lens Equation 1 𝑠 + 1 𝑠 ′ = 1 𝑓
1 𝑠 + 1 𝑠 ′ = 1 𝑓
Sign conventions are all-important!
s > 0: Object on side with incoming light
s’ > 0: Image on side with outgoing light
f > 0: Converging lens
Lateral magnification:
𝑚= 𝑦 ′ 𝑦 =− 𝑠 ′ 𝑠

19. Ray-tracing: Converging Lens

20. Ray tracing: Diverging Lens