L5 : Focal Length of Spherical mirror - Ray Optics, Physics, Class 12 Video Lecture

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still part 4 before going ahead with
part 5 now let us talk about the focal
length of a spherical mirror so right
now we are studying all the terms and
all the concepts related to a spherical
mirror only after that we will talk
about how reflection takes place in a
spherical mirror so what is focal length
of a spherical mirror so till now we
understood what this principal focus
right so focal length is nothing nothing
but the distance of the principal focus
from the pole of the mirror for example
let us consider the case of concave
mirror we saw that in case of a concave
mirror this is how the principal focus
was this is the principal focus right
and this is the pole of the mirror so
what is focal length the distance
between the focus and the pole so this
is my focal length like this distance
would be the focal length now similarly
in case of a convex mirror what happens
in case of a convex mirror this is my
principal focus so the distance of the
focus from the pole that means this
distance is your focal length so focal
length is generally denoted by a small F
so please remember this compliment these
letters which we use to denote these
terms because that is that will also
help you because everywhere you use the
same abbreviations for example focal
length will use a small F for principal
focus we use a capital F for pole we use
a capital P so please get used to all
these abbreviations now we will look at
the relationship between focal length
and radius of curvature so I explain
both the things so do you remember what
is focal length focal length is nothing
but the distance from the principal
focus to the poll that is small egg and
what is the radius of curvature it is
nothing but the radius of the sphere of
which the mirror is a part so this is
capital R so we will find out we will
find out a relationship between the
spoken length and radius of curvature so
this is a very important relationship
actually so what does the Sade's of
curvature it is nothing but you can see
that for example let me consider your
concave mirror this is the concave
mirror let us suppose this is pole let
us say this is my principal focus and
this is the center of curvature so that
means this distance is the radius of
curvature and this distance is the focal
length so we want to find out a
relationship between this focal length
and this radius of curvature like you
got a better idea now that what we are
what are we trying to do so now in order
to derive a relationship between R and F
let us consider this spherical mirror
let us suppose we have a concave mirror
and this is the pole of the mirror this
is the principal focus and this is the
center of curvature let us up and this
is my principal axis of course right let
us suppose there is a ray of light which
is incident on the mirror parallel to
the principal axis so what happens after
reflection it passes through the
principal focus because we saw in our
previous light ray that the rays of
light which were parallel to the
principal axis when they fall on the
mirror after reflection they all met at
the principal focus so we can say that
this ray of light which is parallel to
principal axis when it falls on the
mirror it gets reflected and passes
through the principal focus right let us
say that this ray makes an angle for
example same to theta
so if this angle is to theta this entire
angle will also be to theta
correct okay so now let us name these
points let us call this point as a let
us draw or complete this triangle let us
call this point as B so now if you look
at the triangle if you look at the
strangle e be F what do you see you see
time to theta is equal to a B divided by
B F because tan theta is equal to
perpendicular by base so in this
triangle you see perpendicular is a B
and bases B F so you can write like this
now since we have to involve so we want
to find out a relationship between focal
length which is this F and radius of
curvature which is this eye right so we
have to involve this length as well so
let us draw or let us join this point C
with this point a so what happens this
entire angle goes to theta so now this
is Theta and again this is Theta like so
now let us consider this triangle so if
we consider so if this angle is Theta
what would be this angle because you see
this is also theta so this theta and
this theta are alternate angles right
this theta this theta and this theta
they are alternate angles so this is
also theta so now let us consider
triangle a B C so in triangle ABC you
can say tan theta is equal to
perpendicular that is a B divided by
base that is BC right now here in this
case we can see that theta is a very
small angle right because right now we
are talking about the angle which the
ray of light is making when it falls on
a mirror so this angles are all small
angles now when theta is very small
theta is very small
means that the incident rate is making
small angle with the principal asks
right so if theta is very small in that
case we can say that tan theta can be
written as theta right
similarly tan 2 theta can be written as
2 theta so what happens to these two
equations so you can write these two
equations like this you can say that 2
theta is equal to a B by B F and theta
is equal to a B by this right so this is
your first equation and this is your
second equation so from these 2
equations you can say that a B frumps
from this equation it becomes a B is
equal to 2 theta into B F and from this
equation you get a B is equal to theta
into BC that means if you see the
left-hand side of both these equations
are same therefore that I can decide
should also be equal so 2 theta into B F
should be equal to MU theta into BC so
this theta and theta will cancel so to
be F will be equal to BC now what is V F
B F is nothing but C this being is very
close to the pole that is P so we assume
that B P is very very small since B is
very very close to P so what do we
assume we assume that B P is extremely
small and it can be neglected
therefore this B F can be considered as
PF right so we can say that 2 into F is
equal to what is VC VC is nothing but T
C so this is equal to I so that means
the radius of curvature is equal to
twice the focal length of a spherical
mirror so this is a very important
relation
and you will see that it will be used
every now and then so the rate what this
is and this is true for any spherical
mirror the radius of curvature will
always be twice the focal length of the
spherical mirror so you saw that how did
we derive this relationship today so now
here we have assumed the case of a
concave mirror so instead if you want
instead of concave mirror you can
consider the convex mirror and derive
the same again
the only difference would be that in
here what happened the ray of light
which was incident it passed through the
focus in that case the ray of light will
get diverged and it will appear to pass
through the focus so I will just draw it
for your reference in case of convex
mirror in case of a convex mirror it
would be somewhat like this so a ray of
light incident parallel to the principal
axis after reflection it will diverge in
this way but if you extrapolate it then
it will pass through the principal focus
so in this case we will assume that the
angle so here we assume that it makes an
angle to theta so here also we will
assume that it makes an angle to theta
so here also we will do the same thing
we'll consider this triangle and then
again there will be a centre of
curvature so we will join it with the
centre of curvature also so we do the
same things as we did in the case of
concave mirror right so the calculation
part will remain the same just as well
your diagram will change because this is
a diverging mirror now right so I hope
now you will be able to just for your
own practice you try to derive this
expression fraud a convex mirror as well
which I hope it is clear to you now okay
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