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Current Electricity

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Current Electricity PDF Notes, Important Questions and Formulas

Current Electricity

1. Definition of Current, Current Density, Drift Velocity

  •  Conductor

    In some materials, the outer electrons of each atoms or molecules are only weakly bound to it. These electrons are almost free to move throughout the body of the material and are called free electrons. They are also known as conduction electrons. When such a material is placed in an electric field, the free electrons move in a direction opposite to the field. Such materials are called conductors.

 

  • Insulator:

    Another class of materials is called insulators in which all the electrons are tightly bound to their respective atoms or molecules. Effectively, there are no free electrons. When such a material is placed in an electric field, the electrons may slightly shift opposite to the field but they can't leave their parent atoms or molecules and hence can't move through long distances. Such materials are also called dielectrics.

 

  • Semiconductor:

    In semiconductors, the behavior is like an insulator at low levels of temperature. But at higher temperatures, a small number of electrons are able to free them and they respond to the applied electric field. As the number of free electrons in a semiconductor is much smaller then that in a conductor, its behaviour is in between a conductor and an insulator and hence, the name semiconductor. A freed electron in a semiconductor leaves a vacancy in its normal bound position. These vacancies also help in conduction.

 

 2. ELECTRIC CURRENT AND CURRENT DENSITY

  • When there is a transfer of change from one side of an area to the other, we say that there is an electric current through the area. If the moving charge are positive, the current is in the direction of motion, if they are negative, the current is opposite to the direction of motion, if they are negative, the current is opposite to the direction of motion. If a charge ∆Q crosses an area in time ∆t, we define the average electric current through the area during this time as


    begin mathsize 12px style straight i equals ΔQ over Δt end style

    The current at time t is  begin mathsize 12px style straight i equals limit as Δt rightwards arrow 0 of ΔQ over Δt equals dQ over dt end style

    Thus, electric current through an area is the rate of transfer of charge from one side of the area to the other. The SI unit of current is ampere. If one coulomb of charge crosses an area in one second, the current is one ampere. It is one of the seven base units accepted in SI.

 

3. DRIFT SPEED

  • A conductor contains a large number of loosely bound electrons which we call free electrons or conduction electrons. The remaining material is a collection of relatively heavy positive ions which we call lattice. These ions keep on vibrating about their mean positions. The average amplitude depends on the temperature. Occasionally, a free electron collides or interacts in some other fashion with the lattice.

    The speed and direction of the electron changes randomly at each such event. As a result, the electron moves in a zig-zag path. As there is a large number of free electrons moving in random directions, the number of electrons crossing an area ∆S form one side very nearly equals the number crossing from the other side in any given time interval. The electric current through the area is, therefore, zero.

    When there is an electric field inside the conductor, a force acts on each electron in the direction opposite to the field. The electrons get biased in their random motion in favour of the force. As a result, the electrons drift slowly in this direction. At each collision, the electron starts afresh in a random direction with a random speed but gains an additional velocity v’ due to the electric field. This velocity v’ increases with time and suddenly becomes zero as the electron makes a collision with the lattice and starts afresh with a random velocity. As. the time between successive collisions is small, the electron "slowly and steadily drifts opposite to the applied field (shown figure). If the electron drifts a distance in a long time t, we define drift speed as
    begin mathsize 12px style straight V subscript straight d equals straight l over straight t end style
    If τ be the average time between successive collisions, the distance drifted during this period is


    begin mathsize 12px style straight l equals 1 half straight alpha left parenthesis straight tau right parenthesis squared equals 1 half left parenthesis eE over straight m right parenthesis left parenthesis straight tau right parenthesis squared end style

    The draft speed is begin mathsize 12px style straight V subscript straight d equals straight l over straight tau equals 1 half left parenthesis eE over straight m right parenthesis straight tau end style

    It is proportional to the electric field E and to the average collision-time τ.

    The random motion of free electrons does not contribute to the drift of these electrons. Also, the average collision-time is constant for a given material at a given temperature. We, therefore, make the following assumption for our present purpose of discussing electric current.
    When no electric field exists in a conductor, the free electron stay at rest (V= 0) and when a field E exists, they move with a constant velocity


    begin mathsize 12px style straight V subscript straight d equals fraction numerator eτ over denominator 2 straight m end fraction straight E equals KE text                                 end text. ... text (1) end text end style

    Opposite to the field. The constant K depends on the material of the conductor and its temperature.

    Let us now find the relation between the current density and the drift speed. Consider a cylindrical conductor of cross-sectional area A in which an electric field E exists. Consider a length Vd ∆t of the conductor (figure shown). The volume of this portion is Avd∆t. If there are n free electron per unit volume of the wire, the number of free electrons in this portion is nAVd∆t.  All these electrons cross the area A in time ∆t. Thus, the charge crossing this area in time ∆t is

    ∆Q=n AV∆t e

    Or,  begin mathsize 12px style straight i equals ΔQ over Δt equals nAV subscript straight d straight e end style

    And   begin mathsize 12px style straight j equals straight i over straight A equals neV subscript straight d end style             …. (2)
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