Energy Stored in Capacitor Equation

Introduction to Energy Stored in a Capacitor Equation

Capacitance:

Different conductors have different capacities to hold electric charge. The capacity of a conductor to hold charge depends upon the shape, size and surroundings of the conductor. The capacity of a conductor to hold charge is called capacitance.I like to share this paper capacitor with you all through my article.

For an isolated conductor when we increase the charge on the conductor, its potential also gets increased. The charge on the conductor is directly proportional to the potential of the conductor.

Q a V

Q = CV

Here C is proportionality constant called as capacitance. So capacitance is

C = Q / V

Unit of capacitance is coulomb / volt or farad.


Energy Stored in a Capacitor Equation:

Consider an uncharged capacitor of capacitance C. The capacitor is charged to a potential V when connected to a battery. Let the charge on the capacitor is Q. Suppose at any intermediate state of charging, let the instantaneous charge on the capacitor be q and potential difference across the capacitor is

dV = q /C. The work done to increase the charge dq is given as

dW = dV.dq

= (q /C) dq

The total work done to charge the capacitance from q = 0 to final charge

q = Q is i. e. is the energy stored in capacitor.

W = ? (q/C) dq

= (1/C) (Q2/2)

= Q2 /2C ……….. (1)

= (CV)2 /2C

U = (1/2)CV2 ……….. (2)

Equation (1) & (2) are the relation for energy stored in capacitor.


Effect of Dielectric on Energy Stored in a Capacitor on Equation:

1) When a charging battery is removed from the capacitor and dielectric of constant k is added, then the energy stored in the capacitor decreases by k times.

2) When a dielectric of constant k is added keeping the charging battery connected to the capacitor, then the energy stored in the capacitor increases by k times.

Electric Flux Density

Introduction to electric flux density:

The electric field creates a force on a charge and hence the charge moves along a certain path called the electric flux line. Also the force between two charges acts along a certain path. This path is also called the electric flux line. The electric flux through a surface held inside an electric field represents the total number of electric lines of force crossing the surface in a direction normal to the surface. it is denoted by  ?.

Magnitude of flux depends only on the charge from which it originates. The flux lines are equal to the charge in Coulombs. It is only an imaginary line.  Its direction is same as that of the electric field. It is a scalar quantity.

Electric Flux Lines

Electric flux lines from a point charge:

Electric flux lines between a positive charge and a negative charge:

Electric Flux Density:

1. Electric flux density is defined as the electric flux crossing the surface area.

Mathematically, Electric flux density,  D is defined as

D = ??/ ?S  ,   C/m2

Where, ? is electric flux crossing the differential area, ?S.

The direction of ?S is always outward, normal to ?S, that is, ?S = ?S an .

2. Electric flux density is also defined as (in a general medium)

D = eE , C/m2

Where e is permittivity, F/m

E is electric field strength, V/m

3. Electric flux density in a dielectric medium is given by

D = e0E + P


Where P is polarization of medium

e0 is permittivity of free space.

4. Electric flux density is a vector quantity.

5. The unit of electric flux density is C/m2.

6. In free space, D is in the direction of E.

7. D, in a Gaussian surface, is determined from gauss’s law.

8. D is independent of the medium. 

Heating Effects of Electric Current

Introduction to heating effects of electric current:

When we apply potential difference accross two ends of wire, an electric is set up in the wire.  Such a current is due to motion of free electrons in the wire.  During the motion of electrons they collide with each other and also with ions in the wire.  Due to these collisions kinetic energy of electrons decreases.  This loss in kinetic energy appears as heat and temperature of wire rises.  The amount of heat produced depends on (1) the current passing through the wire  (2) the resistance of the wire and 3) time for which current is passed.
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The amount of heat produced in a conductor is given by the formula H =I2Rt  Joules, where I is the current flowing through the conductor, R is the resistance of the conductor and t is the time for which the current passes through the conductor.

This mathematical equation can be stated in the form of a law which is known as Joule's law.

It states that " the quantity of heat generated (H) in a conductor of resistance (R), when current (I) flows through for time (t) is directly proportional to

i)  the square of the current

ii) the resistance of the conductor

iii) the time for which the current flows.

Applications of Heating Effect of Electric Current:

The heating effect of electric current has many practical applications.  It is used in many domestic appliances such as an electric heater, an electric iron, a geyser, lectric oven etc.

1)  An electric bulb contains a thin filament of metal like tungsten.  It has high melting point.  When current passes through the filament it is heated to high temperature and emits light.  This is the principle on which electric incandescent light bulb works.

2)  An lectric iron used for ironing clothes consists of a coil of high resistance covered by insulating mica sheets and kept inside heavy metal block.  When electric current passes through the coil it gets heated.  The iron metal block gets heated and can be used for ironing clothes.

3)  An important application of heating effect of electric current is a safety device known as "fuse".  Fuse is usuaaly made up of alloys of lead and tin.  It has very low melting point.  It melts with small rise in temperature.  Its diameter is such that it melts when a current passing through it exceeds certain value.  When excess current passes through fuse due to some accident such as short circuit, the wire melts and circuit immediately breaks down.  When shorcircuit occurs, high current flows through the circuit, the fuse wire gets heated up and melts.  The cicuit is broken and current stops flowing.


4) In industry soldering, welding, cutting, drilling and working of electric furnaces are based on heating of electric current.

5) In surgery, a fine heated platinum wire is used for cutting tissues much more efficiently than knife.