Focal Length of Spherical Mirrors Video Lecture | Physics for Airmen Group X - Airforce X Y / Indian Navy SSR

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so now we will talk about focal length
of a spherical mirror what is focal
length of a mirror it is the distance of
the principal focus from pole of the
mirror so now let us look at both the
types of mirror so which mirror is this
so by by the time I finish this lesson
you should not should have no confusion
about which is a concave and which is
upon which mirror so here if you see
this is your concave mirror right
because the inside is sill good so you
for your concave mirror the rays of
light after reflection actually neat at
this point so this is my principal focus
which is my both the center of this
reflective surface that is this is full
so this distance between principal focus
and pole is the focal length that means
FP is the focal length focal length is
generally denoted by a small F so in
this case this is my pole and which is
the focus this is the focus so in this
case also this P F becomes the focal
length right so this is the focal length
of a spherical mirror so what are the
terms related to a spherical mirror
whole center of curvature principal
focus principal axis focal length of a
measured radius of curvature of a mirror
right so these are the terms which have
learned which we learnt so far so now
let us look at the relationship between
focal length and radius of curvature
that means there is a relationship
between F and R so let us try to see the
relationship between F and are the
relationship between focal length and
radius of curvature is that radius of
curvature is equal to twice the focal
length in case of a spherical mirror so
this relationship is true for both
concave mirror as well as convex mirrors
so when we consider a concave mirror
it is something like this is my pole
this is my principal focus and this is
my center of curvature so what is odd
odd is this distance right that is the
radius of the sphere door tells my focal
length that is this distance F so it
sees that R is equal to twice of the
focal length now if you want to prove it
we can also prove this so let us try to
prove this relationship so let us prove
it for concave mirror however it will it
will be true for both concave and convex
and you can prove it for both of them
but for now I am considering a concave
mirror and we prove this relationship
for a concave mirror it is of course
this is the pole of the middle this is
the principal focus and this is the
center of curvature it is because this
is an incident ray when it reaches when
when it strikes the mirror it gets
reflected and then a reflected ray then
passes this way there is not a ray which
passes through
this centre of curvature
now let us suppose that this angle is
Theta this angle is also theta so what
will be this angle if you look at it
this makes a Z so this angle and this
angle will be alternate angles so this
will also be theta and what will be this
angle this entire angle should be equal
to this angle that is alternate angles
again so this should be equal to 2 theta
right so now let us name these two
points sighs let me call this point as a
and this point is already P and let us
suppose if I draw a perpendicular here
and all that as B just for my
construction purpose so now if you look
at this figure you can see from the
triangle ABC if you look at triangle a B
C you can say that from your
trigonometry you can say that tan theta
is equal to perpendicular by base so
what is perpendicular that is a B and
what is the base that is BC so tan theta
is equal to a B by D C now you look at
triangle a B F so in triangle ADF you
can say that tan 2 theta is equal to a B
by FB that is this is equal to a B by B
F right now here this angle theta is
assumed to be very small because
incident rays
make very small angle to the principal
axis because incident rays don't weigh
very large angles so we can see that
theta is quite small
now if theta is very small we can say
that tan theta is approximately equal to
theta this is all these things you would
have studied in your trigonometry right
so I am NOT going into the detail that
why tan theta becomes theta and theta
and small all this for all these things
you can refer to your mathematics majors
so tan theta becomes theta so using that
these two expressions will become theta
is equal to a B by B C that will become
my first equation and 2 theta will be
equal to a B by B if this will become my
second equation right so from these two
equations you can say that a B is equal
to 3 times 2 BC here a B is equal to 2
theta 2 B F therefore we can say that
from equation 1 and equation 2 we can
say that theta into BC is equal to 2
theta into D F right so theta theta will
get cancelled therefore D C is equal to
2 into B F what is BC if you look at it
what is BC this PB this length PB is
extremely small right when compared to
this distance BC or bf b P is a nice one
so if DP is very small we can see that
BC is nothing but PC take since it is
very small I mean let us suppose if you
have a length like this and if you add a
very small length so this length is
almost equal to this length because the
difference is very negligible so you can
write DC as PC similarly you can write
DF as PF now what is PC PC it is nothing
but radius of curvature that is the
distance between the pole P and center
of curvature C and what is P f PF is
nothing but the distance between pole
and principal focus that is focal length
so therefore you can say ad is equal to
2 into L that is the radius of curvature
is twice the focal length of a spherical
mirror so this is how we proved it so
and you have understood the relationship
between focal length and radius of
curvature now this relation holds true
for a convex mirror as well if you want
to improve it in a similar way for
convex mirror also thank you please
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