Current and Current Density

Current and Current Density

• The current is defined as the rate of flow of charge and is measured in amperes.

Key Point : A current of 1 ampere is said to be flowing across the surface when a charge of one coulomb is passing across the surface in one second.

• The current is considered to be the motion of the positive charges. The conventional current is due to the flow of electrons, which are negatively charged. Hence the direction of conventional current is assumed to be opposite to the direction of flow of the electrons.

• The current which exists in the conductors, due to the drifting of electrons, under the influence of the applied voltage is called drift current.

• While in dielectrics, there can be flow of charges, under the influence of the electric field intensity. Such a current is called the displacement current or convection current. The current flowing across the capacitor, through the dielectric separating its plates is an example of the convection current.

• The analysis of such currents, in the field theory is based on defining a current density at a point in the field.

• The current density is a vector quantity associated with the current and denoted as .

Key Point : The current density is defined as the current passing through the unit surface area, when the surface is held normal to the direction of the current.

• The current density is measured in amperes per square metres (A /m²).

1. Relation between I and 

• Consider a surface S and I is the current passing through the surface. The direction of current is normal to the surface S and hence direction of  is also normal to the surface S.

• Consider an incremental surface area dS as shown in the Fig. 5.2.1 (a) and  is the unit vector normal to the incremental surface dS.

• Then the differential current di passing through the differential surface dS is given by the dot product of the current density vector 

• But if  is not normal to the differential area  then the total current is obtained by integrating the incremental current which is dot product of  over the surface . This is shown in the Fig. 5.2.1 (b). Thus in general,

• Thus if is in A /m² and   is in m2 then the current obtained is in amperes (A). It may be noted that J need not be uniform over S and S need not be a plane surface.

2. Relation between  and p v

• The set of charged particles give rise to a charge density ρv in a volume v. The current density  can be related to the velocity with which the volume charge density i.e. charged particles in volume v crosses the surface S at a point. This is shown in the Fig. 5.2.2. The velocity with which the charge is getting transferred is  m/s. It is a vector quantity.

• To derive the relation between  and ρv, consider differential volume ΔV having charge density ρv as

shown in the Fig. 5.2.3. The elementary charge that volume carries is,

ΔQ = ρv Δv … (5.2.6)

Let ΔL is the incremental length while ΔS is the incremental surface area hence incremental volume is,

Δv = ΔS ΔL ... (5.2.7)

ΔQ = ρv ΔS ΔL    ... (5.2.8)

• Let the charge is moving in x direction with velocity  and thus velocity has only x component vx.

Note : Velocity is denoted by small italic letter while the volume is denoted by small normal letter.

• In the time interval Δt the element of charge has moved through distance Δx, in x direction as shown in the Fig. 5.2.3. The direction is normal to the surface ΔS and hence resultant current can be expressed as,

ΔI = ΔQ / Δt ... (5.2.9)

But now, ΔQ = ρv ΔS Δx  as the charge corresponding the length Δx is moved and responsible for the current.    

ΔI = ρv ΔS (Δx / Δt) ... (5.2.10)

• Such a current is called convection current and the current density is called convection current density.

Key Point : The convection current density is linearly proportional to the charge density and the velocity with which the charge is transferred.

Ex. 5.2.1 In cylindrical co-ordinates,  Find the current crossing through the region 0.01 ≤ r ≤ 0.02 and intersection of this region with the ϕ = constant plane.

Sol. : The current is given by integral form of the continuity equation as,

Ex. 5.2.2 If the current density inside a box bounded by five planes, x = 0, y = 0, y = 2, z = 0 and 3x + z = 3 is . Find the total current flowing out of the surfaces of the box.

Sol . :

Ex. 5.2.3 Find the total current in a circular conductor of radius 4 mm if the current density varies according to J (104/r) A/m².

Sol. : The current is given by,

Examples for Practice

Ex. 5.2.4 Find the total current in outward direction from a cube of 1 m, with one comer at the origin and edges parallel to the co-ordinate axes if 

[Ans. : 3 A]

Ex. 5.2.5 A current density  in the spherical co-ordinate system. a) How much crrent flow through the spherical cap r = 3 m, 0 < θ <π/6, 0 < ϕ < 2 π .

b) The same total current as found in (a) flows through the spherical cap r = 10 m, 0 < ϕ < a, 0 < ϕ > < 2π. What should be the value of a ?

[Ans.: 70.6858 A, 28.47° or 0.4969 rad]

Ex. 5.2.6 If  Calculate the current passing through a hemisphere shell of radius 20 cm.

[Ans.: 31.416 A]

Ex. 5.2.7 In a cylindrical conductor of radius 2 mm, the current density varies with the distance from the axis according to J = 10³ e^-400 r A/m². Find the total current.

[Ans.: 7.5115 mA]

Ex. 5.2.8 Find the current crossing the portion of y = 0 plane defined by  where   is the current density.

[Ans.: 4 mA]

Review Questions

1. Derive the relation between I and .

2. Derive the relation between  and pv.