Section-II EM-I (1)

8/7/2019 Section-II EM-I (1)

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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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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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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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