Electrical currents

Imagine that a wire is connected to a battery and a resistor as shown in Figure 2-22a. The battery is a chemical source that provides particles with a positive charge and a negative charge. In the metallic wire, the ions are stationary and a portion of the electrons are free to move. Benjamin Franklin gave us the convention that the direction of the flow of the current, however should be in the direction of the motion of the positive particles. Between the two terminals of the battery, there will be an electric field that will accelerate the electrons. Since the conductivity of a wire is significantly greater than the conductivity of the surrounding air, the motion of these accelerated electrons will follow the path of the wire. The resulting current in this circuit is called the

“conduction current.” When we later describe electric fields that depend upon time, we will encounter another current that is called a “displacement current.”

The current that passes through the wire in Figure 2-22b can be computed from Ohm's law. Since we are more interested in local effects rather than in global effects, we would rather relate this current to a current density

A J I and

a voltage difference 'V across an incremental length 'L, i.e. an electric field. The resistance R of the wire is given by

A R L

V

' where V is the conductivity of the

wire. This definition of resistance follows from the intuition gained in circuits where the total resistance of a circuit is computed by adding the resistors in series (i. e. increased length of the wire ¨L) and the conductors in parallel ( i. e.

increased cross-sectional area A). The current density in a wire is therefore defined using Ohm’s law

RA E L L V A

R V A

J I ' V

'

¸ '

¹

¨ ·

©

§ '

(2.101)

This is a generalization of Ohm's law. One Ampere of current at a point is defined as the passage of one Coulomb of charge passing this point in one second. We are assuming that none of the parameters that appear in (2.101) depend upon the magnitude of any of the other parameters which could be found in nonlinear materials. One could think of a resistor whose resistance would change with increasing values of current caused by a heating of the resistor. Such nonlinear effects are important in practice but will not be considered here.

An alternative derivation for the conductivity V follows from the definition of the current density J = U_vv_drift where Uv is the electron volume charge density and v_drift is an average electron drift velocity. The drift velocity is proportional to the electric field E, the proportionality constant being called the mobility P of the material. Hence, we write

J = U_v v_drift = U_v PE = V E (2.102) where the conductivity V = U_v P.

Figure 2-22. (a) A simple electrical circuit consisting of a battery and a resistor.

(b) A current I flows through the wire whose area is equal to A = Sa² where a is the radius of the wire.

The total current I that passes through the wire is computed from the inte-gral of the current density J integrated over the cross sectional area A of the

(a) (b)

V

R

I

A

'L

wire. The current density J is a vector since it has both a magnitude and a direction

³

I J ds (2.103)

If the current is uniformly distributed in a cylindrical wire whose radius is a, this integral can be easily performed and we find that the total current I that passes through the wire is given by

I = J Sa² (2.104)

This is equal to the product of the current density times the cross-sectional area of the wire. However, if the current is nonuniformly distributed in the wire, this integration requires more care as will be shown with an example. Later, we will encounter cases where the current is constrained to flow just on the surface of an object. These currents are called surface currents. Having now presented some fundamental definitions for the currents in terms of local current densities and cross sectional areas, we are prepared to explore various properties of magnetic fields that will be created by these currents.

Example 2-20. Given an inhomogeneous current density J = (3y²zu_x - 2x³zu_y + zu_z) A/m²,

find the total current passing through a square surface at x = 1 in the u_x direction.

The dimensions of the surface are 1 ” y ” 2, 1 ” z ” 2.

Answer: The differential surface area is defined as ds = dydz ux. Therefore, the scalar product on the current density and the current density will yield a current only in the ux direction. We write

Example 2-21. Calculate the current that flows through a wire whose radius is a.

The inhomogeneous current density in the wire is uz

Nonuniform currents can be important in high frequency applications in which one encounters “skin effects.”

Answer: The current is calculated from the following integration:

³

The power that is dissipated within a conducting material can be calculated from the electric field and the current density that we have just encountered. The power density in a particular volume is defined as

p = J x E (W/m³) (2.105)

The total power that is absorbed within the volume is calculated by integrating (2.105) over the entire volume 'v. This power is converted into another form and it is given the name “Joule heating.” It is measured in SI in units

Watts = Joules / seconds.

The reader has probably experienced the warming effects of Joule heating in cooking a meal on an electric stove or being warmed on a cold winter’s night by an electric heater that is present in the room. Suffice it to say, this is a very important effect that has many practical applications.

Example 2-22. Calculate the power that is dissipated within a resistor that has a uniform conductivity V. The voltage between the two ends of the resistor is V and a current I passes through the resistor.

Answer: From (2.105), we obtain the power density. The total power loss is calculated from the integration of the power density over the volume

VI