wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

What are conjugate foci? Deduce the following relationship between the focal length f, of a spherical mirror, distance of object of u and distance of image v. 1u+1v=1f.

Open in App
Solution

The sets of points on the principal axis of a lens where the position of the object and the image can be interchanged are called conjugate foci.

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 ΔABC
ABC=A'B'C'(each 90o)
ACB=A'CB'(vertically opposite angles)
Hence, ABAB=CBBC ........(1)
In ΔDEF and ΔA'B'F
DEF=A'B'F(each 90o)
DFE=A'FB'(vertically opposite angles)
Hence, DEAB=EFFB
But; DE=AB
ABAB=EFFB ......(2)
From equation (1) and (2)
CBBC=EFFB
If the aperture of mirror is very small i.e., point E is very near to point P.
Then, EF=PF
CBBC=PFFB
PBPCPCPB=PFPBPF ......(3)
Applying sign convention, PB=U,PC=R=2f
PB=y,PF=f
Put in equation (3)
U+2f2f+v=fv(f)
U+2f2f+v=fv+f
(u+2f)(v+f)=f(2f+v)
uvuf2fv+2f2=2f2fv
uv=uf+2fvfv
uv=uf+fv
Dividing uvf, we get
uvuvf=ufuvf+fvuvf
1f=1v+1u
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 90o)
ACB=A'CB'(Common angles)
Hence,
ABAB=BCBC .....(1)
In ΔDEF and ΔA'B'F
DEF=A'B'F(each 90o)
DEF=A'FB'(Common angles)
DEAB=EFBF ......(2)
From equation (1) and (2)
BCBC=PFBF
If the aperture of mirror is very small i.e., point E is very near to point P then EF=PF(approx.)
BCBC=PFBF
PB+PCPCPB=PFPFPB ........(3)
Applying sign convention,
PB=u,PC=+R=+2f
PB=+v,PF=+f
Put in equation (3),
U+2f2fv=ffv
(u+2f)(fv)=f(2fv)
uf+uv+2f2=2f2fv
uv=uf+2fvfv
uv=uf+fv
Dividing uvf, we get
uvuvf=ufuvf+fvuvf
1f=1v+1u
This is the mirror formula for convex mirror.

666819_628994_ans_ee0e7d41d53a4d98a4ee7a93f552a67b.png

flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Image Shift in a Glass Slab
PHYSICS
Watch in App
Join BYJU'S Learning Program
CrossIcon