CURRENT DENSITY: ELECTRICAL CONDUCTION AND EXAMPLES - PHYSICAL - 2025

It is called current density to the amount of current per unit area through a conductor. It is a vector quantity, and its modulus is given by the quotient between the instantaneous current I that passes through the cross section of the conductor and its area S, so that:

Said like this, the units in the International System for the current density vector are amps per square meter: A / m ². In vector form the current density is:

The current density vector. Source: Wikimedia Commons.

Current density and current intensity are related, although the former is a vector and the latter is not. The current is not a vector despite having magnitude and meaning, since having a preferential direction in space is not necessary to establish the concept.

However, the electric field that is established inside the conductor is a vector, and it is related to the current. Intuitively it is understood that the field is stronger when the current is also stronger, but the cross-sectional area of ​​the conductor also plays a determining role in this regard.

Electric conduction model

In a piece of neutral conducting wire like the one shown in Figure 3, cylindrical in shape, the charge carriers move randomly in any direction. Inside the conductor, according to the type of substance with which it is made, there will be n charge carriers per unit volume. This n should not be confused with the normal vector perpendicular to the conducting surface.

A piece of cylindrical conductor shows current carriers moving in different directions. Source: self made.

The proposed conducting material model consists of a fixed ionic lattice and a gas of electrons, which are the current carriers, although they are represented here with a + sign, since this is the convention for current.

What happens when the conductor is connected to a battery?

Then a potential difference is established between the ends of the conductor, thanks to a source that is responsible for doing the work: the battery.

A simple circuit shows a battery that by means of conductive wires lights a light bulb. Source: self made.

Thanks to this potential difference, the current carriers accelerate and march in a more orderly way than when the material was neutral. In this way he is able to turn on the bulb of the circuit shown.

In this case, an electric field has been created inside the conductor that accelerates the electrons. Of course, their path is not free: although the electrons have acceleration, as they collide with the crystalline lattice, they give up some of their energy and disperse all the time. The overall result is that they move a little more orderly within the material, but their progress is certainly very little.

As they collide with the crystalline lattice they set it to vibrate, resulting in heating of the conductor. This is an effect that is easily noticed: conductive wires become hot when they are passed through by an electrical current.

Crawling speed

Current carriers now have a global motion in the same direction as the electric field. That global speed they have is called the drag speed or drift speed and is symbolized as v _d.

Once a potential difference is established, the current carriers have a more orderly movement. Source: self made.

It can be calculated by means of some simple considerations: the distance traveled inside the conductor by each particle, in a time interval dt is v _d. dt. As stated before, there are n particles per unit volume, the volume being the product of the cross-sectional area A and the distance traveled:

If each particle has charge q, what amount of charge dQ passes through area A in a time interval dt ?:

The instantaneous current is just dQ / dt, therefore:

When the charge is positive, v _d is in the same direction as E and J. If the charge were negative, v _d is opposite the field E, but J and E still have the same direction. On the other hand, although the current is the same throughout the circuit, the current density does not necessarily remain unchanged. For example, it is smaller in the battery, whose cross-sectional area is larger than in the thinner conductor wires.

Conductivity of a material

It can be thought that the charge carriers, moving inside the conductor and continuously colliding with the crystalline lattice, face a force that opposes their advance, a kind of friction or dissipative force F _d that is proportional to the average speed that carry, that is, the drag speed:

F _d ∝ v

F _d = α. v _d

It is the Drude-Lorentz model, created at the beginning of the 20th century to explain the movement of current carriers inside a conductor. It does not take quantum effects into account. α is the constant of proportionality, whose value is in accordance with the characteristics of the material.

If the drag speed is constant, the sum of forces acting on a current carrier is zero. The other force is that exerted by the electric field, whose magnitude is Fe = qE:

The entrainment velocity can be expressed in terms of the current density, if it is properly solved:

From where:

The constants n, q and α are grouped in a single call σ, so that finally we obtain:

Ohm's law

The current density is directly proportional to the electric field established inside the conductor. This result is known as Ohm's law in microscopic form or local Ohm's law.

The value of σ = nq ² / α is a constant that depends on the material. It is about electrical conductivity or simply conductivity. Their values ​​are tabulated for many materials and their units in the International System are amps / volt x meter (A / Vm), although there are other units, for example S / m (siemens per meter).

Not all materials comply with this law. Those that do are known as ohmic materials.

In a substance with high conductivity it is easy to establish an electric field, while in another with low conductivity it takes more work. Examples of materials with high conductivity are: graphene, silver, copper and gold.

Application examples

-Resolved example 1

Find the entrainment velocity of free electrons in a copper wire of cross-sectional area 2 mm ² when a current of 3 A passes through it. Copper has 1 conduction electron for each atom.

Data: Avogadro's number = 6.023 10 ²³ particles per mole; electron charge -1.6 x 10 ^-19 C; density of copper 8960 kg / m ³; molecular weight of copper: 63.55 g / mol.

Solution

From J = qnv _d the magnitude of the drag velocity is cleared:

How come the lights come on instantly?

This speed is surprisingly small, but you have to remember that cargo carriers are continually colliding and bouncing inside the driver, so they are not expected to go too fast. It may take an electron almost an hour to go from the car battery to the headlight bulb for example.

Fortunately, you don't have to wait that long to turn on the lights. One electron in the battery quickly pushes the others inside the conductor, and thus the electric field is established very quickly as it is an electromagnetic wave. It is the disturbance that propagates within the wire.

The electrons manage to jump at the speed of light from one atom to the adjacent one and the current begins to flow in the same way that water does through a hose. The drops at the beginning of the hose are not the same as at the outlet, but it is still water.

- Worked Example 2

The figure shows two connected wires, made of the same material. The current that enters from the left to the thinnest portion is 2 A. There the entrainment speed of the electrons is 8.2 x 10 ^-4 m / s. Assuming that the value of the current remains constant, find the entrainment velocity of the electrons in the portion to the right, in m / s.

Solution

In the thinnest section: J ₁ = nq v _d1 = I / A ₁

And in the thickest section: J ₂ = nq v _d2 = I / A ₂

The current is the same for both sections, as well as n and q, therefore:

References

1. Resnick, R. 1992. Physics. Third expanded edition in Spanish. Volume 2. Compañía Editorial Continental SA de CV
2. Sears, Zemansky. 2016. University Physics with Modern Physics. 14 ^th. Ed. Volume 2. 817-820.
3. Serway, R., Jewett, J. 2009. Physics for Science and Engineering with Modern Physics. 7th Edition. Volume 2. Cengage Learning. 752-775.
4. Sevilla University. Department of Applied Physics III. Density and intensity of current. Recovered from: us.es
5. Walker, J. 2008. Physics. 4th Ed. Pearson. 725-728.