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Millikan Experiment
The Mathematical Part
The Physical Basics
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In the Millikan Experiment, The basic physical concepts
used are Force due to Electrical Field and The Air
Resistance
We know that the force on a charged particle in an
electrical field is given by
Where F is the force experienced, e is the charge and E is
the Electric Field Intensity
The Physical Basics
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The Air resistance on any spherical particle is given by
“Stokes Law” which states that the Frictional Force on
any spherical body is directly proportional to the velocity
of the body.
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Here the constant of Proportionality is given by gamma
and is called the coefficient of air resistance.
Now … The Experiment
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This was the setup that was used by Millikan.
In the Experiment
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Millikan used a fine spray of ionized oil droplets, which he
allowed to be acted on by gravity but to which he also
applied an electric field in the direction opposite gravity,
i.e. up.
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By tuning the electric field, he balanced the force due to
the pull of gravity and the electric field force so that the
drops remained suspended in space.
In the Experiment
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Thus, if a drop has a positive charge and a mass , the
force balance condition becomes
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where is the magnitude of the electric field. Solving for
the charge-to-mass ratio, we have
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Now The Electric Field was Switched off
Electric Field is Switched Off
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In this case, there are still two forces acting on the drops
One is the gravitational force and the other is the
frictional force due to air resistance
However, when the drops are allowed to fall, the total
force is , since friction opposes gravitational force.
Electric Field is Switched Off
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Now By Newton’s Second Law of Motion
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From this equation we can solve this v as
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Since is very small and is relatively large, the exponential
factor quickly decays to 0, leaving simply a constant velocity
Terminal Velocity
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The Velocity becomes constant and is called the Terminal
Velocity
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This velocity can be measured. Thus M can easily be
calculated.
And Using the e/m ratio, We can calculate the value of
charge.
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Quantization of Charges
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Talking of the basic principle of charges i.e. all objects have a
charge equal to an integral multiple of the fundamental charge
e.
The technical problem to prove this is that it is not known
how many electrons are stripped off each drop before the
experiment is performed.
Thus, each drop will have a different charge q that is a multiple
of the fundamental charge unit e.
By subtracting successive values of q obtained for different
drops, however, it is possible to find a ``smallest'' value, which
can be assumed to be a prediction of e, although the possibility
that the result is still a multiple of e cannot be ruled out.