Potential Energy of a System of Charges (Part 2) Video Lecture | Physics for EmSAT Achieve

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potential energy of a system of end
point charges where you have multiple
point charges now we looked at a system
of two charges now let us suppose if we
have a system where we have several
charges so what would be the potential
energy of such a system let us suppose
we start with one by one charge let us
suppose I have a charge q1 which I bring
from infinity to some point in let us
call this point as a so let us suppose I
bring this charge q1 from infinity to
this point a right so what happens what
is the work done to bring this charge q1
from infinity to a the work done is
equal to 0 because there is no electric
field which exists at all right because
before q1 there was no other charge
present there so there will be no
electric field there will be no electric
potential due to any other charge
particle therefore the work done to
bring q1 from infinity to a is equal to
zero because there was no charge
particle at all and therefore there was
no electric field at all now let us
suppose I bring another charge particle
q2 to some point say B let us call this
point as D so what is the work done when
we bring this charge Q 2 that is work
done to bring q2 from infinity to B so
what will be the work done in this case
now remember friends this work done is
basically this is nothing by definition
of potential energy so we are basically
finding out potential energy like this
so we're going to bring q2 from infinity
to B now when q2 comes from infinity to
B as soon as it comes to point B
an electric potential exists at point B
due to the charge q1 right so this is
similar to the system of two charges
which we discussed in the previous slide
so in this case what happens this work
then becomes equal to the charge q2 into
the potential difference that is the
potential at B minus potential at
infinity so this becomes equal to q2
into potential at be because this
potential at infinity is zero so this
becomes equal to q2 into potential at B
is due to charge q1 right so this
becomes K q1 divided by r1 to let us
suppose this distances are 1/2 so this
becomes K q1 q2 divided by r1 2 so this
is the work done to bring q2 from
infinity to B right so what is this this
is basically net now this was nothing
but potential energy u 1 similarly this
is nothing but potential energy u 2
similarly let us calculate work done to
bring a charge Q 3 from infinity to some
point C now you have two charges already
present now you bring in a third charge
Q 3 so you bring this charge to some
point C and let us suppose this
distances are 1 3 that is a distance of
Q 3 from Q 1 is R prime 3 and the
distance of Q 3 from Q 2 is R 2 3 right
so in this case what what happens as
soon as this charge Q 3 comes at Point C
so the work then becomes equal to
magnitude of the charge Q 3 into the
potential difference that is potential
at C minus potential at infinity so this
is zero therefore this becomes equal to
Q 3 into potential at Point C now the
potential at Point C is due to both of
these char
that is there is a potential at Point C
due to q1 also there is a potential at
Point C due to q2 right so we can write
this as potential at C do a plus
potential at C due to be right so this
will become your potential at this point
right so what would be the potential at
C due to a that would be nothing but K
q1 divided by r1 3 similarly what would
be the potential at C due to be that
would be k q2 divided by r3 right so
your YouTube will become so from this
you can say so you - I'm sorry you three
will become K q1 q3 / add 1/3 + k q2 q3
divided by r2 3 right similarly let us
calculate the value of u4 and u5 you 6
and so on and as you keep bringing more
and more charge particles you keep
getting more and more terms like this so
therefore God would be the total
potential energy of the system that will
be equal to the sum of these potential
energies right so u 1 is 0 what is u2 u2
was K q1 q2 divided by r1 2 what is Q 3
Q 3 was K Q 1 Q 3 divided by r1 3 + K Q
2 Q 3 divided by r2 3 similarly when you
calculate u 3
when you when you calculate you fool you
get something like this K Q 1 Q 4
divided by r1 4 + K Q 1 Q 3 divided by
r1 3 + K Q 1 I'm sorry this is K Q 2 Q 4
divided by r2 4 + K Q 3 Q 4 divided by
r3 4 like so this is how it will keep
continuing so what can you write we can
write this in this term if we want to
write it in in general form so we can
say that the total potential energy is
equal to summation of K Q I QJ / ad IG
for all pairs but now if you look at
this carefully you see that Q 1 Q 2 if
it should be such that I and J should
never be equal and also the term should
not repeat that means for each pair of I
and J let us suppose if I say I is 1 and
J is 2 so for I is equal to 1 and J is
equal to 2 only 1 term should come it
should not repeat but if we write it
like this then it will repeat right once
it will be Q 1 Q 2 KQ 1 Q 2 by R 1 2
again you will get another term K Q 2 Q
1 by r 2 1 right now if I write it in
general like this many people make
mistake that they write this expression
as this which is not correct because all
terms are not included in this
expression if you see you will never
find the term which says K Q 2 Q 1 by r
2 1 there is no such term like this and
kind of thumb will never come so for
each pair of I and J only one term
should come that means only this is to
be noted only one term for each pair of
IJ therefore how do we write this we
write it as half that means the total
number of term which we will get like
this you have this if you have this that
means you are excluding those to placate
pairs so you can write it as half into
summation of K Q I QJ / R IJ such that I
is equal to 1 to n J is also equal to 1
to n but I is not equal to J so this is
your complete expression or this is a
complete general expression for
potential energy of a system of n point
charges I hope you understood why I
multiplied it by half because if you
want to write this expression this term
whatever you are getting since this is
an infinite expression so if you want to
write it in a general form you cannot
write it like this it is not appropriate
to write it like this because if you
write it like this
this will include all the pairs but you
want all pairs of IJ but only once you
don't want them to repeat because if you
use this expression what will happen you
will get one value as KQ and Q 2 by R 1
2 which is this again you will get
another term as KQ 2 Q 1 divided by R 2
1 which we don't want therefore we have
multiplied it by half so that we get
only those number of terms which we want
I hope it is clear to you okay so this
is how we calculate potential energy of
a system whether it has 1 charge or it
has multiple charges now if you look at
this diagram here I have drawn I mean I
told you up to charge Q 3 you try to do
it on your own you try
that later suppose q4k so what happens
when q4 comes like this you can
calculate I am leaving to calculate u4
and u5 to you this is your assignment
try to calculate on your own so when you
try to calculate Q for you I assume that
this distance is add one for you assume
that the distance from q3 is at three
for you assume that the distance of q4
from q2 is add to 4 and then again you
proceed similarly similarly if you get
another charge Q 5 say again you try to
calculate the potential energy
experienced when this charge comes in so
and this process keeps continuing as
many number of charges keeps coming into
the system right so that thank you
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