Electric-Current Density

Electric current defined as

is the most popular parameter describing current in everyday life. In many technical and scientific descriptions the current density is used. The current density is the ratio of current intensity to the area, perpendicular to current direction, through which this current is flowing. Let's look at the Fig.1, which shows a conductor build from two rods of different diameters together with cross sections of these rods.

Fig.1 Current in conductor with changing area of cross section

The current I is the same at any point of this conductor. This is a consequence of the fact that the number of electric charges passing the cross section of this conductor at any point must be the same. Charges cannot leave the conductor between any chosen points and cannot be “produced” as well. To better understand this statement imagine that water is flowing through a tube of the shape like the one in Fig.1. In the unit of time the same amount of water passes through any cross section of this tube. Otherwise there should be a leak of water between observed cross sections or some water would have to be added to the tube between these points.
For the values of the current I and area S₁ and S₂ given in Fig.1 we will have two values of current density J₁ = 4.0/ 4.0 = 1 A/cm²   and  J₂= 4.0 / 1.0 = 4.0 A/ cm² In other words, the current density is the intensity of current passing through the unit area of the surface perpendicular to the direction of the flow of this current. This definition holds true for a uniformly distributed flow of charges.

There are many situations, where the flow of charges is not uniformly distributed across the cross section of the conductor. The classic example of such a situation is a high frequency current in a metallic wire. Later on you will learn that the current flows mainly in a very thin outer layer of this wire. It is a so called skin effect. The higher the frequency of the current, the thinner the layer actually conducting the current.

The mathematical definition of current density, which is applicable to any possible distribution of charges flowing in the conductor is

where is the current density at the area element , and I is the total current through area. The “arrow notation” for vectors is used here.

If you are not familiar with calculus and the vector representation of area, and you don’t know what the meaning of an integral is, leave this definition alone and remember only, that current density is current per unit area as mentioned at the beginning of the paragraph. The SI unit of this quantity is Ampere / square meter (A/m²).

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