Define principal focus and focal length
Scenario
Kagaz jalane wala experiment yaad hai? Sun ki light ek magnifying glass se pass karke ek chhoti si bright dot banate the aur paper jal jata tha.
Woh saari dhoop (sunlight) ek choti si dot par kaise akatthi (collect) ho gayi?
Mental model
Focus woh milan bindu (meeting point) hai jahan parallel light rays girti hain.
Jab saari parallel light rays mirror se takrakar ek fixed point par aakar milti hain, toh use Principal Focus kehte hain. Us point se mirror ke centre tak ki doori Focal Length hoti hai.
Explanation
Spherical mirrors ka ek specific point hota hai jahan aane wali light focus hoti hai. Jab light rays principal axis (mirror ke centre se nikalne wali seedhi line) ke parallel aati hain, toh reflection ke baad wo ek specific point se guzarati hain (ya guzarti hui lagti hain). Is special point ko Principal Focus (`F`) kehte hain.
Concave mirror ke case mein, parallel rays sach mein aakar ek point par milti hain (converge hoti hain), isliye iska focus 'real' hota hai aur mirror ke front mein hota hai. Convex mirror mein, rays fail jati hain (diverge) par jab hum unhe peeche extend karte hain, toh lagta hai jaise ek hi point se nikal rahi hain. Isliye convex ka focus 'virtual' aur mirror ke peeche hota hai.
Mirror ke ekdum centre point ko Pole (`P`) bolte hain. Pole (`P`) se lekar Principal Focus (`F`) tak ki distance ko Focal Length (`f`) kehte hain. Focal length depend karti hai mirror ke curve par; jitna jyada curved mirror hoga, focal length utni hi choti hogi.
Ek aur zaroori relation yaad rakhna hai: Focal length (`f`), Radius of Curvature (`R`) ki aadhi hoti hai. Yani `f = R / 2`. Agar sphere bada hai jiska mirror hissa hai, toh radius zyada hoga aur focal length bhi badi hogi.
Worked example
Problem: The radius of curvature of a spherical mirror is `20 cm`. What is its focal length?
- Focal length aur radius of curvature ka formula lagayein.
- R ki value substitute karein.
Answer: `f = 10 cm`
Problem: A concave mirror has a focal length of `15 cm`. If we consider the sphere of which this mirror is a part, what is its radius?
- Radius of curvature focal length ka dugna hota hai.
- Focal length ki value daalein.
Answer: `R = 30 cm`
Key points
- Principal Focus (F): The point on the principal axis where rays parallel to it converge or appear to diverge from after reflection.
- Focal Length (f): The distance between the pole of the mirror (`P`) and the principal focus (`F`).
- Relation with Radius: `f = R / 2`. Focal length is half the radius of curvature.
Common mistakes
- Confusing Pole and Focus: Pole mirror ki surface par hota hai, jabki focus hawa mein mirror ke aage (concave) ya peeche (convex) hota hai. Focal length in dono ke beech ki doori hai.
Recall questions
- What is the relationship between the radius of curvature (`R`) and focal length (`f`) of a spherical mirror?
- Does a convex mirror have a real or virtual principal focus?
- What is the principal focus of a convex lens?
Questions & answers
Find the focal length of a convex mirror whose radius of curvature is `32 cm`.
`16 cm`
Approach: `f = R / 2 = 32 / 2 = 16 cm`.
Define the focal length of a spherical mirror.
The distance between the pole and the principal focus.
Approach: It is the direct definition from the sign convention layout.
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