Deduce the following relationship between the focal length f of a spherical mirror distance of object u and distance of image v
1/u+1/v=1/f
Answers
Explanation:
For concave mirror:
Let MPN=Concave mirror
P=Pole
F=Principle focus
C=Centre of curvature
PC=Principle axis
AB=Principle acis
A;B'=Real image of object AB formed by concave mirror
DE=Perpendicular drawn from D on the principle axis
In ΔABC and ΔA
′
B
′
C
′
∠ABC=∠A'B'C'(each 90
o
)
∠ACB=∠A'CB'(vertically opposite angles)
Hence,
A
′
B
′
AB
=
B
′
C
CB
........(1)
In ΔDEF and ΔA'B'F
∠DEF=A'B'F(each 90
o
)
∠DFE=A'FB'(vertically opposite angles)
Hence,
A
′
B
′
DE
=
FB
′
EF
But; DE=AB
∴
A
′
B
′
AB
=
FB
′
EF
......(2)
From equation (1) and (2)
B
′
C
CB
=
F
′
B
′
EF
If the aperture of mirror is very small i.e., point E is very near to point P.
Then, EF=PF
∴
B
′
C
CB
=
FB
′
PF
PC−PB
′
PB−PC
=
PB
′
−PF
PF
......(3)
Applying sign convention, PB=−U,PC=−R=−2f
PB
′
=−y,PF=−f
Put in equation (3)
−2f+v
−U+2f
=
−v−(−f)
−f
−2f+v
−U+2f
=
−v+f
−f
(−u+2f)(−v+f)=−f(−2f+v)
uv−uf−2fv+2f
2
=2f
2
−fv
uv=uf+2fv−fv
uv=uf+fv
Dividing uvf, we get
uvf
uv
=
uvf
uf
+
uvf
fv
f
1
=
v
1
+
u
1
This is the mirror formula for concave mirror.
(2) For convex mirror:
Let, MPN=convex mirror
P=Pole
F=Principle focus
C=Centre of curvature
PC=Principle axis
AB=Object
A'B'=Image of object AB formed by convex mirror
DE=Perpendicular drawn from point D on the principle axis.
In ΔABC and ΔA'B'C
∠ABC=∠A'B'C(each 90
o
)
∠ACB=∠A'CB'(Common angles)
Hence,
A
′
B
′
AB
=
B
′
C
BC
.....(1)
In ΔDEF and ΔA'B'F
∠DEF=∠A'B'F(each 90
o
)
∠DEF=∠A'FB'(Common angles)
A
′
B
′
DE
=
B
′
F
EF
......(2)
From equation (1) and (2)
B
′
C
BC
=
B
′
F
PF
If the aperture of mirror is very small i.e., point E is very near to point P then EF=PF(approx.)
B
′
C
BC
=
B
′
F
PF
PC−PB
′
PB+PC
=
PF−PB
′
PF
........(3)
Applying sign convention,
PB=−u,PC=+R=+2f
PB
′
=+v,PF=+f
Put in equation (3),
2f−v
−U+2f
=
f−v
f
(−u+2f)(f−v)=f(2f−v)
−uf+uv+2f
2
=2f
2
−fv
uv=uf+2fv−fv
uv=uf+fv
Dividing uvf, we get
uvf
uv
=
uvf
uf
+
uvf
fv
f
1
=
v
1
+
u
1
This is the mirror formula for convex mirror.
solution
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