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- 1. Steady Electric CurrentsTopics covered in this chapter are: • Current Density and Ohms Law • Electromagnetic force and Kirchoffs Voltage Law, Continuity Equation and Kirchoffs Current Law. • Power Dissipation and Joules law • Boundary Conditions for Current DensityIntroduction: In our discussion so far we have considered field problems that are associated with static charges. In thischapter we consider the situation when the charges can be in motion and thereby constituting current flow. Due to the movement of free charges, several types of electric current can be caused:Conduction Current is due to the drift of electrons and/or holes and occurs in conductor and semiconductor.Motion of ions gives rise to electrolytic currents and convection current results from motion of electrons and/orions in an insulating medium such as liquid, rarified gas and vacuum. It is worth mentioning here that in time varying scenario, bound charges give rise to another type of currentknown asDisplacement current, which we shall consider in more detail in later chapters.Current Density and Ohms Law:In our earlier discussion we have mentioned that, conductors have free electrons that moverandomly under thermal agitation. In the absence of an external electric field, the averagethermal velocity on a microscopic scale is zero and so is the net current in the conductor. Underthe influence of an applied field, additional velocity is superimposed on the random velocities.While the external field accelerates the electron in a direction opposite to it, the collision withatomic lattice however provide the frictional mechanism by which the electrons lose some of themomentum gained between the collisions. As a result, the electrons move with someaverage drift velocity . This drift velocity can be related to the applied electric field by therelationship ......................(3.1)where is the average time between the collisions.
- 2. The quantity i.e., the the drift velocity per unit applied field is called the mobility ofelectrons and denoted by .Thus , e is the magnitude of the electronic charge and , as the electron driftsopposite to the applied field.Let us consider a conductor under the influence of an external electric field. If represents thenumber of electrons per unit volume, then the charge crossing an area that is normal tothe direction of the drift velocity is given by: ........................................(3.2)This flow of charge constitutes a current across , which is given by, ................(3.3)The conduction current density can therefore be expressed as .................................(3.4)where is called the conductivity. In vector form, we can write, ..........................................................(3.5)The above equation is the alternate way of expressing Ohms law and this relationship is valid ata point.For semiconductor material, current flow is both due to electrons and holes (however in practice,it the electron which moves), we can write ......................(3.6) and are respectively the density and mobility of holes.The point form Ohms law can be used to derive the form of Ohms law used in circuit theoryrelating the current through a conductor to the voltage across the conductor.Let us consider a homogeneous conductor of conductivity , length L and having a constantcross section S as shown the figure 3.1. A potential difference of V is applied across theconductor.
- 3. Fig 3.1: Homogeneous ConductorFor the conductor under consideration we can write, V = EL ..................................(3.7)Considering the current to be uniformly distributed, .............(3.8)From the above two equations, ............................(3.9)Therefore, ............(3.10)where is the resistivity in and R is the resistance in .Electromotive force and Kirchhoff’s Voltage LawFrom our earlier discussion we know that ......................(3.11)Using the point form of Ohm’s Law we can write ......................(3.12)
- 4. Thus we observe that an electrostatic field cannot maintain a steady current in a closed circuit.Motion of charged carriers in a circuit, which is required to establish a steady current, is adissipative process while charges moving in a closed path in a conservative electrostatic fieldneither gain nor lose energy after completing one trip round the circuit. The loss of energy isnormally supplied by sources of non-conservative field (e.g. battery, generator, photovoltaic celletc) and provides a driving force for the carriers. Fig 3.2: Typical BatteryIf we consider the battery shown in figure 3.2, chemical action causes accumulation of positiveand negative charges, these charges establish electrostatic field both inside and outside thebattery. When the battery is under open circuited condition no current flows through it and thenet force acting on the charges must be zero. Therefore, ......................(3.13) Fig 3.3: Battery in Closed Circuit When we have a closed circuit as shown in figure 3.3, we must have at all points ......................(3.14)
- 5. , being a conservative field, . Since is zero outside the battery and consideringfinite conductivity only in the conductor region, ......................(3.15)We define, Electromotive force or EMF to be . In a circuit where we have multiplesources and multiple resistors, ..........................................(3.16)The expression (3.16) is that of Kirchhoff’s voltage Law, which states that algebraic summationof the EMF’s (voltage rise) in a circuit is equal to the algebraic sum of the of the voltage dropacross the resistors.Continuity Equation and Kirchhoff’s Current LawLet us consider a volume V bounded by a surface S. A net charge Q exists within this region.If a net current I flows across the surface out of this region, from the principle ofconservation of charge this current can be equated to the time rate of decrease of chargewithin this volume. Similarly, if a net current flows into the region, the charge in the volumemust increase at a rate equal to the current. Thus we can write, .....................................(3.17) or, ......................(3.18)Applying divergence theorem we can write, .....................(3.19)It may be noted that, since in general may be a function of space and time, partialderivatives are used. Further, the equation holds regardless of the choice of volume V , theintegrands must be equal.Therefore we can write,
- 6. ................(3.20)The equation (3.20) is called the continuity equation, which relates the divergence of currentdensity vector to the rate of change of charge density at a point.For steady current flowing in a region, we have ......................(3.21)Considering a region bounded by a closed surface, ..................(3.22)which can be written as, ......................(3.23)when we consider the close surface essentially encloses a junction of an electrical circuit.The above equation is the Kirchhoff’s current law of circuit theory, which states that algebraicsum of all the currents flowing out of a junction in an electric circuit, is zero.

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