The reflecting surface sends light back to the left. So the focus and centre of curvature of a concave mirror lie on the same side as the object (left) → against incident light → negative.
Concave mirror: f<0, R<0.
Convex mirror: focus is behind the mirror (right side) → with incident-light direction → positive: f>0, R>0.
Real image forms in front (left) → v<0; virtual image behind → v>0.
Light passes through, continuing left→right. For a convex (converging) lens, the second focus where rays converge is on the right → with light → positive: f>0. For a concave (diverging) lens: f<0.
Convex lens: f>0. Concave lens: f<0.
Real image forms on the right → v>0; virtual image on the left (same side as object) → v<0.
Using paraxial geometry (small angles), for a concave mirror with magnitudes a=∣u∣,b=∣v∣,F=∣f∣:
a1+b1=F1
Now apply signs: object left ⇒u=−a, real image left ⇒v=−b, concave focus left ⇒f=−F. Substitute:
−v1+−u1?
Multiply the magnitude relation by −1:
−a1−b1=−F1⟹v1+u1=f1
because −a=u,−b=v,−F=f. So the signed mirror formula is:
v1+u1=f1Why this step? We replaced each magnitude with its signed coordinate so the single equation now holds for any configuration without redrawing.
What is the chosen positive direction in the New Cartesian convention?
The direction of incident light (left → right).
Sign of object distance u for a normal real object?
Negative (u<0), because the object lies opposite to incident light.
Focal length sign of a concave mirror?
Negative (f<0).
Focal length sign of a convex mirror?
Positive (f>0), focus is behind the mirror.
Focal length sign of a convex lens?
Positive (f>0).
Focal length sign of a concave lens?
Negative (f<0).
Mirror equation (signed)?
v1+u1=f1.
Lens equation (signed)?
v1−u1=f1.
Magnification for a mirror?
m=−v/u.
Magnification for a lens?
m=+v/u.
What does a negative magnification tell you?
The image is inverted.
For a mirror, what does v<0 mean physically?
Real image forming in front of the mirror (on the object's side).
Why is the lens formula "minus" while the mirror's is "plus"?
Light transmits through a lens (object & image on opposite sides) vs reflects off a mirror (same side).
Recall Feynman: explain it to a 12-year-old
Imagine you and the light are walking down a hallway. Wherever the light is heading — straight ahead — we call "plus." Wherever it came from — behind you — we call "minus." A toy you put behind you (the object) gets a minus number. A mirror bounces the light backward, so a mirror's special "focus" point lands behind the mirror sometimes (plus) or in front (minus). A magnifying glass lets light keep going forward, so its focus is ahead — plus. We just label everything with these plus/minus tags, and then one little formula tells us where the picture appears and whether it's upside-down. The tags are the whole trick!
Dekho, sign convention basically ek "rule book" hai jisse pata chalta hai kaunsi distance plus hai aur kaunsi minus. New Cartesian convention bolta hai: jis taraf light jaa rahi hai (left se right), wo plus direction hai. Object hamesha left mein hota hai, light ke against, isliye object distance u hamesha minus hota hai. Ye ek baat yaad rakho to aadha kaam ho gaya.
Ab focus ka sign element ki geometry se aata hai, "converging hai ya diverging" se nahi. Concave mirror ka focus mirror ke saamne (left, light ke against) hota hai → isliye f<0. Convex mirror ka focus peeche (right) → f>0. Lens mein light cross karke aage jaati hai, to convex lens ka focus right mein → f>0, aur concave lens ka f<0. Sabse common galti yahi hai ki bachche concave mirror ko bhi f>0 maan lete hain kyunki wo converging hai — wo galat hai, geometry dekho.
Formula bhi yaad rakho: mirror = plus (v1+u1=f1), lens = minus (v1−u1=f1). Reason simple hai — mirror light ko wapas bhejta hai (object aur image same side), lens transmit karta hai (opposite sides). Jab answer aata hai, uska sign khud bata deta hai image real hai ya virtual, seedhi hai ya ulti. Isliye signs ko mat ignore karo — wahi aapko free mein nature aur orientation bata dete hain. Practice karte waqt har problem mein pehle saare signs likho, phir substitute karo — marks pakke.
