The Mirror Equation Video Lecture | Physics Class 12 - NEET

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reflection and refraction pattern is
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back nine before going ahead read part
10 let us now talk about mirror equation
what is mirror equation it is an
equation which relates image distance
with object distance and focal length
that means the dist MN as you have seen
before also that as the distance of the
object changes the position of the image
also changes right so that means there
has to be a relationship between the
image distance object distance and the
focal length so as the focal length of
the lens change focal length of the
mirror changes or the distance of the
objects changes your image will also
change its position so this is the
relationship which connects all these
three now as a matter of convention will
denote image distance by V object
distance by u and focal length as F
right so what does my mirror equation
mirror equation is the mirror equation
goes something like this it says that 1
by u plus 1 by V is equal to 1 by F so
this is the middle equation which
relates the image distance with object
distance and focal length now let us try
to prove this mirror equation
mathematically I mean how did we get
this with our equation so let us
consider a concave mirror here and let
us consider any object let us suppose we
have I mean any arbitrary position we
have considered for now let us suppose
we have some object a B now let us try
to plot the ray diagram and let try to
plot its image so let us suppose this is
a ray which goes parallel to the
principal axis after reflection it will
pass through the principal focus there
is another way which passes through the
center of curvature and after reflection
it goes back along the same path it is
suppose I have to
another rule it falls obliquely towards
the pole and therefore it goes back
obliquely as well so all these theories
they meet up at this point and therefore
we get a real inverted image here so
this B - a - is a real and inverted
image which we get so now if we look at
this diagram and if we consider these
strangles now first first of all let me
name things here this is a this is B let
me call this as M and this is already a
- B - f and triangle MPF so this is MTF
the strangle and this is a - B - f so if
you compare these two triangles you have
vertically opposite angles you have this
as 90 degree and these as 90 degree so
we can say that since all that angles of
these two triangles are similar so you
can see that triangles A - B - f and
triangles a MPF add similar triangles
now by property of similar triangles we
can say that B - a - divided by p.m.
that means B - a - divided by p.m. will
be equal to B - f divided by F T you can
draw these two triangles separately -
somewhat like this this is B - this is a
- this is f this is M and this is P so
you can say that B - a - by MP or p.m.
is equal to D - f divided by F P that is
by property of similar triangles like so
this we can write it as B - a - divided
by p.m. now what is p.m. p.m. is nothing
but a B if you look at this diagram the
a be the same as being ever so you can
write it as B - a - divided by a B is
equal to B - f divided by F P so let me
call this equation as equation one right
now looking at this diagram we can say
that the angle this angle and this angle
will remain the same I have not drawn it
very nicely I mean the oblique angle
with the pole will be equal to this
angle with which it is getting reflected
that means if this angle is angle one
this angle is angle two with an angle
one is equal to angle 2 so therefore we
can say that angle triangles a B P so a
B P will be similar to triangle a - V -
P so here also let me draw these two
triangles separately it is suppose this
is triangle a B P this is my angle one
so this will make an angle with P and
there is neither triangle like this and
this is B - this is a - and this is
angle - right this is how I have drawn
it separately so if you compare these
two triangles and these two angles are
similar this is 90 degree this is 90
degree so these angles will also be
equal so by a in similarity these two
triangles will also be similar
therefore by property of similarity in
this case also we can say that a B
divided by B - a - is equal to PB
divided by PB - so let me call this
equation as equation 2 so if you compare
equation 1 and equation 2 you see that
the right hand side has got similar
values right so we can say that from
equation 1 and equation 2 we can say B -
f divided by a
P is equal to PB - divided by PB now
from this we can sit now we can see it
see that if you look at this figure you
can see that B - f what is B - if this
is B - f what does B - f it is nothing
but P D - - PF right we can say that
since B - f is equal to P B - minus P F
so if we substitute this value we get P
B - - PF / p f is equal to PB - divided
by PB right now if you look at the
figure what is my object distance object
distance is the distance at which the
object is located the object is located
at this distance so the distance PD is
the object distance image is formed at B
- so the distance PB - is the image
distance so the object distance is PB
which is denoted by U and what would be
the sign the sign would be negative
because it is towards the left hand side
of the pole what would be the image
distance now what is the image distance
the image distance as I mentioned is PB
- which is denoted by V with a negative
sign because as per the sign Convention
rules both U and V are on the left hand
side of the pole right and what is P f
PF is nothing but the focal length which
is f that is also negative because it
also falls on the left hand side of the
pole now putting these values in this
equation we get minus V Plus F divided
by minus F is equal to minus V by minus
u so this becomes V minus F divided by F
is
equal to V by you so we get V by F minus
one is equal to V by u now dividing this
entire equation by V so dividing the
entire equation by V on both left hand
side and right hand side what do we get
we get 1 by F minus 1 by V is equal to 1
by u or we can write it as 1 by u plus 1
by V is equal to 1 by F so this equation
relates the image distance object
distance and the focal length of the
mirror now these derivations might not
be very important from your examination
point of view but I derived it just to
make you understand that how do how did
we get these expressions right because
it doesn't make any point to memorize
things without knowing its origin okay
so now we have shown the mirror equation
for concave mirrors now let us have a
comparison of the middle equation for
concave mirror and convex mirror however
this the reader equation which I
mentioned just now is a generalized
mirror equation which will hold true for
both concave and convex we did provide
this you consider the sign convention by
putting values for UV and F right but
now I'll show you if for concave mirror
you wrote 1 by V plus 1 by U is equal to
1 by F considering sign convention what
will be the mirror equation for a convex
mirror what will happen in case of a
convex mirror if I assume the object to
be some object say a V how would the ray
diagram look like anyway of light
parallel to principal axis after
reflection will appear to pass through
this principal focus another way of
light which falls obliquely to pole gets
deflected obliquely so these two rays
meet at meet virtually at some point
like this so an image a - B - cells form
so in this case what where is my focal
length my focal length is P F so in this
case P F is equal to F positive in this
case what is my image distance that is P
B so in this case P B is equal to you in
negative because it is on the left side
of the pole what about the image
distance it is P V - which is V positive
right in concave mirror UV f everything
was negative but in case of convex
method here you see that image and focal
length they are on on the right hand
side of port so they are positive so if
you put these values in the equation in
our previous slide that is if you put
all these values in this equation which
I had mentioned let me call this
equation as equation a then what happens
so equation a becomes if you put these
values in equation a it becomes V minus
F divided by F is equal to plus V
divided by minus u so when I solve this
and then again you get V by F minus 1 is
equal to minus V by u so dividing
throughout by V you get 1 by V minus 1
by U is equal to 1 by F so considering
the sign Convention for upon Beck's
mirror you see that the mirror equation
carries a negative sign with you however
by solving problems do not confuse
yourself with different formula you
consider this equation as the
generalized equation for mirror equation
doesn't matter whether the inner is a
concave or convex mirror but by putting
the values of UV and F you incorporate
sign convention that means if it is a
concave mirror take V as negative if it
is a convex mirror take V as positive
and so on you will understand
clearly when we go ahead and solve some
problems as well thank you please visit
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