Potential Energy in an External Field Video Lecture | Physics for JEE Main & Advanced

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electrostatic potential part 19 is
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still part 18 before going ahead with
part 19 an about potential energy of a
system of charges now let us look at
potential energy in an external field
let us suppose if you have a charged
particle and if you place that charged
particle in an external field and in
that field is not produced by the
charged particle that field is produced
by some other particle or some other
source which we do not know we just know
that there is an electric field which is
already existing and you put a charged
particle or you put a point charge in
that electric field so what will happen
to the potential energy of that point
charge so let us suppose that these
field lines represent an external
electric field let us suppose this is an
external electric field and we have a
point charge say Q 1 which we bring from
infinity to some point C P let us call
this point as P so we bring this charged
particle from infinity to some point P
which is in the electric field of some
external sources and we do not know what
which Force produce this electric field
it is some external electric field so
how do we calculate the potential energy
of this charged particle that is Q 1 in
this external electric field so here we
assume two things firstly so one
assumption is so what are our
assumptions here one assumption is the
standard assumption that potential at
infinity is equal to zero the second
assumption is that the electric field of
charge Q does not bother this electric
field electric field of Q one doesn't
bother the external electric field V
right that means in a way we are
assuming that this Q one is a small
charge which doesn't have a prominently
high
field of its own so we are neglecting
the electric field of charge q1 so now
we say that this is the external
electric field e so corresponding to
this electric field there will be an
electric potential let us say B so let
us calculate the work done to bring q1
from infinity to point-p so what would
be the work done that will be equal to
the magnitude of the charge into
potential difference that is potential
at P minus potential at infinity now
this is 0 so this becomes q1 into
potential at point-p
now what would be the potential at point
B P the potential at Point P will be
equal to let us suppose some value B
which is the potential corresponding to
this electric field E and that external
electric field is e so corresponding to
that electric field there will be some
potential so let us call that potential
as B so this will be equal to V
therefore now let us say that this V so
what what can be say what was this this
was nothing but potential energy so we
can say that potential energy is equal
to the magnitude of the charge times the
potential at that point this potential
is a function of R because the potential
at different points will be different so
this potential V of R denotes the
potential at Point P which is denoted by
R from the origin I mean R represents
the position vector of point P let us
suppose if I say that let us assume I
consider this as my coordinate axis let
us suppose this is my origin so the
position vector of this point P is
denoted by I so it says that the
potential at any point R multiplied by
the charge gives the potential energy at
any point in an external electric field
on a single point charge right now let
us look at the potential energy in an
external field due to assist
ten of two charges now let us suppose
you have two charges say Q one and Q so
what would be the potential energy now
now let us look at it so again in this
case also let us of course we bring
there are two charges Q 1 and Q 2 so let
us suppose first we bring Q 1 from
infinity to some point say P let us call
this point as P so what would be u 1 u 1
is the potential energy of charge Q 1
that is work done to bring Q 1 from
infinity to point-p so this will be
equal to magnitude of q1 into potential
difference that is potential at t minus
potential at infinity potential at
infinity is zero therefore this becomes
Q 1 into potential at P so this becomes
equal to q1 into potential at ad 1 let
us suppose that we consider this as the
coordinates right
and later suppose we denote the position
of this and P point P a by R 1
so we say that this will be equal to the
potential energy of the first factor now
let us calculate the potential of the
energy of the second particle so what
would be the work done to bring Q 2 from
infinity to point Q let us call this
point as Q so what will be the work done
in this case this case also it would be
the charge Q 2 into the potential
difference that is potential at Q minus
potential at infinity this is again
equal to 0 so this will be Q 2 into
potential which is a function of R 2
length but
in this case there is also a work done
to bring from infinity to p2 now when
you bring in this case there is another
term which is involved when you
calculate the potential at Q there is
another potential due to this charge q1
right so when you see this this VQ is
because of this that is the external
field plus there is the potential at Q
do to P right so we will consider that
as well so it becomes D of R 2 plus K
into Q 1 divided by later suppose these
distances are 1 2 so this is R 1 2 so
this becomes the potential energy for
the second charge therefore what is the
net potential energy therefore the net
potential energy is equal to u 1 plus u
2 that is equal to Q 1 into V of R 1
plus Q 2 into V of R 2 plus K into Q 1
into Q 2 divided by I 1 2 so this is the
net potential energy of a system of two
charges in an external field so these
values come from the external field and
these values come from the other charges
which are present in the system right
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