Post on 08-Apr-2018
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Lecture Notes Capacitor to Circuits
By. S.M Alay-e-Abbas
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The definition of Capacitance
Capacitor is a device that stores electrical energy.
Consider a sphere with charge Q and radius R.
From previous problems we know that the potential at the surface is
Putting more charge on the sphere stores more energy, but the ratio of energy or
potential to the charge depends only on R and not on Q or V. That is
Its true for all charged objects that the ratio of energy to charge only depends on
the shape of the object this ratio is defined as the capacitance of the object
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Capacitors in a Circuit
Consider the capacitors connected in series
The total voltage V must equal the sum of voltage drops across each capacitor that
is
Using the definition of capacitance
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The law of conservation of charges require that the charge on each capacitor must
equal Q i.e.
Hence the capacitance of the capacitors connected in series becomes
Consider now capacitors connected in parallel
From the law of conservation of charges the sum of the charges on each capacitor
must equal the total charge supplied by the source i.e.
Using the definition of the capacitors we get
From the law of conservation of energy the voltage drop across each capacitor
must be the same
So now the total capacitance of the capacitor must be
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Lecture Notes Capacitor to Circuits
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Energy Stored in a CapacitorSuppose we are trying to put total charge Q on the capacitor. How much energy
will it take to do so?
Assume at some point the charge on the capacitor is q and the total potential
energy is V then the potential energy required to put a small charge dq on the
capacitor will be
Using the definition of capacitance
To find the total energy required to charge the capacitor from q=0 to q=Q we
integrate the above relation to get.
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Lecture Notes Capacitor to Circuits
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We can attribute this energy to the field instead of the capacitor and the volume it
occupies. Using the capacitance of the parallel plate and the relationship between
the field and the potential we can write
In terms of energy density
This gives the energy density of due to electric field inside a capacitor.
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Dielectrics and Capacitors
Dielectrics are insulators. Electrons are not free to flow from one molecule to
another. The atoms in a dielectric can have dipole moments. In a typical chunk of
dielectric material these dipoles are randomly aligned and therefore produce no net
field as shown.
When a dielectric is placed between the plates of a capacitor with a surface charge
density o the resulting electric field, Eo, tends to align the dipoles with the field.
This results in a net charge density i induced on the surfaces of the dielectric
which in turns creates an induced electric field, Ei, in the opposite direction to the
applied field. The total field inside the dielectric is reduced to,
The dielectric constant is defined as the ratio of these two fields
Substituting the value of E and solving for induced field we get
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Note that is pure insulator and is perfect conductor.
To know how the introduction of the dielectric effects the capacitance of the
capacitor we proceed as follows. The potential difference is
Thus the potential difference will be smaller by a factor of 1/K. and applying the
definition of the capacitance we get
The capacitance is lager by a factor of K.
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Current and Resistors
When a potential difference is applied to a conductor an electric field is created
inside. Immediately the free charges begin to flow to cancel the field. It is this flow
of charge that we will study.
Current: The rate at which charge flows.
Now we want to calculate how fast are the electrons moving?
Consider the figure below
The speed of the electrons can be written as
The time can be found by using
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Now the speed becomes
Defining the free electron density and the current density as
we get speed as
This speed is called drift velocity
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Resistivity and Ohms Rule
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Energy Transfer in Circuits
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Circuits
In a circuit, charges move from one place to another carrying energy. These
charges can be thought of as buckets that carry energy around a circuit. The battery
fills the buckets. The buckets are emptied at various places around the circuit, but
the buckets themselves never disappear. They return to the battery to be refilled.
These basic ideas are summarized in Kirchoff's Rules and are applicable to even
the most complicated circuits.
Kirchoffs Rules
The Junction Rule
"The current into any junction is exactly equal to the current out of the junction."
This theorem is explained by the Law of Conservation of Charge.
The Loop Rule
"The sum of all the voltage drops around any loop in a circuit must be zero."
This theorem is explained by The Law of Conservation of Energy and the fact that
the electric force is conservative.
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Resistors In series
The Loop theorem requires
But V =IR
And
Hence
The junction theorem means that
Resistors In parallel
The junction theorem means that
Also
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The loop theorem requires that
Hence
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Electrical Meters
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RC Circuit
Now we use Kirchoffs Rules to analyze circuit containing both R and C
components. Consider the RC circuit shown below
When the switch is closed between a and b the battery starts to charge the
capacitor. When the capacitor is charged the no further current flows. The question
is how long does this takes?
At some intermediate time the current in the circuit is i and the charge is q. by
applying the loop rule we can write
From the definition of current the current must equal the rate at which the capacitor
is being charged
Hence the above equation becomes
This equation can be solved for q(t) by solving dq/dt and integrating.
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Solving for q
The graph below shows the variation of charge with time on the capacitor
It follows that
Now consider the circuit again
When the switch b is connected with c the capacitor starts to discharge. The loop
theorem requires that
The current must equal the rate at which the capacitor discharges i.e i= dq/dt.
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The Equation from the loop rule becomes
Integrating
The graph for the discharge is
Note that
RC is taken as called time constant of the circuit.
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