Expression : Drift Velocity of electron - Current Electricity Video Lecture - Class 12

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us say that initially what did I say
initially the electrons are moving
randomly and then we say that they have
a thermal velocity such that the average
thermal velocity is zero so that means
initially when I say initially I means
before applying the external electric
field initially what happens random
motion okay and such that the average
thermal velocity is equal to zero so let
us suppose that if the average if the
thermal velocity of the electrons are V
1 V 2 V 3 and so on till V n if we
assume that there are n electrons in
that case if these are the values of
thermal velocities then the average
thermal velocity is equal to zero right
now let us suppose that if we apply an
external electric field E then what
happens let us suppose that this is my
conductor
so this is of length L and we see that a
potential difference V exists between
this let us suppose this is my electron
so the electrons mo starts moving
towards this if this is positive and
this is negative so the electron moves
towards the positive end so this V is
nothing but the potential difference
between the ends because this is
positive this is negative so this is
this V represents the potential
difference between the n now we already
know that there is a relationship
between electric field E and the
potential difference V so e can be
written as V divided by L because we
know that electric field is gradient of
potential so e will be equal to V by L
now under the influence of this external
electric field E what happens each
electron experiences a coulombs force
because whenever you apply electric
field whenever the electric field comes
into play coulombs force also comes into
play so each electron experiences
Coulomb force so what does that force
given by that force Coulomb force F is
equal to Q into E so here Q is nothing
but charge on electron so this becomes e
into e so this is the force experienced
by each electron so what would be the
acceleration of each electron
acceleration of electron will be equal
to force per unit mass because force is
equal to mass into acceleration so this
will be equal to e into e divided by m
where m is mass of electron right so
this will be the acceleration of
electron so as soon as electric field is
applied the electron will get strongly
attracted towards the positive terminal
and they will start moving very fast and
they will accelerate so the acceleration
of the electron is given by this
expression that is mine
e into e divided by mass of electron but
the gaining velocity due to this
acceleration is lost during collision of
the electron with the positive ion so
gain in velocity due to acceleration is
lost during collision with positive ions
right that's what I explained you in the
previous slide with positive ions so
what will happen during this collision
the positive ion will also experience a
force but they will not move because
they are heavy because their mass is
much more as compared to electron so
what will happen after the collision
what would be the velocity of an
electron let us consider any one
electron velocity of an electron will be
equal to what initially the velocity of
this electron was v1 let us suppose
right now we are considering the first
electron so initially the velocity was V
1 but now what will be the velocity it
will be nothing but we know that from an
emetic equation v is equal to u plus 80
where u is the initial velocity a is the
acceleration and T is the time so final
velocity will be equal to initial
velocity that is v1 which is the thermal
velocity this was the initial velocity
plus what does acceleration acceleration
is nothing but this acceleration let us
call this acceleration as a so this will
be acceleration a into time taken let us
call this time taken as tau 1 similarly
the velocity of second electron will be
equal to V 2 plus a tau 2 and so on so
on for so on and so forth for all the
other electrons so what will be the
average of these velocities average of
these
velocities will be equal to nothing but
the drift velocity because as I told you
drift velocity is nothing but the
average of the velocity gained by each
of the electron so average of these
velocities will be your drift velocity
letters do not drift velocity as V D
this D denotes drift so drift velocity
will be equal to v1 plus a tau 1 plus v2
plus a tau 2 plus V 3 plus a tau 3 plus
so on divided by how many electrons are
there there are n electrons so divided
by n so this is how we will calculate
the average of drift velocity right
therefore the drift velocity will be
equal to what will be this this will be
nothing but v1 plus v2 plus v3 plus so
on you take all wheels together plus a
into tau 1 plus tau 2 plus tau 3 plus so
on divided by n now this can be written
as V 1 plus V 2 plus V 3 plus dot dot
dot divided by n plus a into tau 1 plus
tau 2 plus tau 3 plus dot dot dot
divided by n now what is this term this
term is nothing but the average of the
thermal velocities of all the electrons
and as we started this expression here I
showed that this is equal to 0 so from
this we can say that this term will be
equal to 0 so we are left with a into
tau where tau is nothing but the average
of the time taken by all the electrons
before it collides with some positive
ion right because what is tau 1 tau 1 is
after what time the electron collides
with the positive ions correct
similarly tau 2 is the time after which
the second electron collides with the
positive ion therefore we say that this
tau is nothing but the average time
collapsed since each electrons suffered
collision so what is this this is
average time collapsed since elections
each electron suffered collision right
therefore we get drift velocity is equal
to a into tau now what is this
acceleration in in the slide we in in
the previous slide we found acceleration
as minus e into e divided by M therefore
the expression for drift velocity is
given by this expression so the drift
velocity is directly proportional to the
electric field which is applied it is
directly proportional to the average
time collapsed between collisions and it
is inversely proportional to the mass of
electron so this is how we calculate
drift velocity for electrons thank you
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