Electric Current and Power

Chapter Learning Objectives: After completing this chapter the student will be able to:  Calculate a line integral  Calculate the total current flowing through a surface given the current density.  Calculate the resistance, current, electric field, and current density in a material knowing the material composition, geometry, and applied voltage.  Calculate the power density and total power when given the current density and electric field.

You can watch the video associated with this chapter at the following link:

Historical Perspective: André-Marie Ampère (1775-1836) was a French physicist and mathematician who did ground-breaking work in electromagnetism. He was equally skilled as an experimentalist and as a theorist. He invented both the solenoid and the telegraph. The SI unit for current, the Ampere (or Amp) is named after him, as well as Ampere’s Law, which is one of the four equations that form the foundation of electromagnetic field theory.

Photo credit: https://commons.wikimedia.org/wiki/File:Ampere_Andre_1825.jpg, [Public domain], via Wikimedia Commons.

10.1 Mathematical Prelude: Line Integrals

As we saw in section 5.1, a surface integral requires that we take the dot product of a vector field with a vector that is perpendicular to a specified surface at every point along the surface, and then to integrate the resulting dot product over the surface. Surface integrals must always be a double integral, since they must involve an integral over a two-dimensional surface. We also have a special symbol for a surface integral over a closed surface, such as a sphere or a cube:

(Copies of Equations 5.1 and 5.2)

While a surface integral requires a double integral to be taken of a dot product over a surface, a line integral requires a single integral to be taken of a dot product over a linear path. The limits of the integration are specified in the equation, but it is also essential that the path to be followed must also be specified, typically in text or a figure to accompany the equation. Equation 10.1 shows the standard notation for a line integral.

(Equation 10.1)

Again, we have a special symbol for a “closed-loop” line integral that begins and ends at the same point (like a race track):

(Equation 10.2)

Here, we do not have to specify the limits of integration, since the beginning and ending of the loop are the same point. Additionally, it doesn’t matter where we begin and end the loop, because the integral will always yield the same result.

Line integrals are especially important in calculations that involve movement against a force over a specified path. In this case, the dot product of force and motion gives instantaneous power required for the motion, and the integral along the path gives the change in potential energy, which can also be represented as work:

(Equation 10.3)

Example 10.1: A particle is subject to the following force at all points in the xy plane:

If it follows a path from point a (1,1) horizontally to point b (4.1) and then from there vertically to point c (4,3), how much work has been done on the particle? y

(4,3) (1,3) d c

a b (1,1) (4,1)

Example 10.2: Consider the particle from Example 10.1. Assume it is still subject to the same force, and it moves from point c (4.3) horizontally to point d (1,3) and then returns to point a (1,1). How much work was done on the particle during this portion of the trip?

We will often break a closed loop down into two or more legs, and we can add the line integral from each leg to find the closed loop integral. In the case of two legs, this can be represented as:

(Equation 10.4)

Example 10.3: Using your answers from Example 10.1 and 10.2, how much work was done on the particle during the entire closed loop path from a to b to c to d and back to a?

You might be surprised to see that this result is not zero. Our experience with the physical world (especially the conservation of energy and Kirchhoff’s Voltage Law) suggest that this closed- loop line integral should be zero. This would be true for “Conservative Fields,” in which all possible closed-loop line integrals are zero. For “Non-Conservative Fields,” at least some closed loop integrals can be non-zero. Most of the fields we will consider in this class are conservative, so your intuition will be helpful in most cases.

Example 10.4: Consider a particle that is subject to the following force at all points along the xy plane:

Using rectangular coordinates, determine the work done on this particle as it moves from point a to point b along a circular path from (5,0) to (0,5). y

R=5

x b

Example 10.5: Repeat Example 10.4 using cylindrical coordinates.

Line integrals will be important to us beginning next chapter when we introduce Ampere’s Law.

10.2 Electric Current and Current Density

As we have previously discussed, electric charges will rearrange themselves to move to a position of lower potential energy. So far, we have discussed very quick changes in charge arrangement that happen only occasionally. This would be similar to knocking over a cup of water and watching as it moves to an arrangement of minimum potential energy (usually all over the floor).

But we can also apply external force on the system that causes the charges to be constantly unsatisfied with their position. This is similar to water constantly flowing over a water wheel, where externally applied force (water flowing from a higher elevation) creates an environment of continuous motion (the wheel turning as long as water is flowing):

Figure 10.1. Water continuously moving toward lower energy causes the wheel to spin (Image credit: https://pixabay.com/en/water-wheel-traditional-technology-312329/ , public domain)

Of course, if the water supply at a higher elevation were to be cut off or dry up, the wheel would stop spinning.

In the same way, an external power source (such as a battery or a nuclear power plant) can cause a continuous flow of electrical current. Notice that as soon as we begin to talk about current flow, we have left the world of electrostatics.

You already know that we use the letter I to represent current, because C is already used for capacitance. (Ampère, the French physicist outlined in this chapter’s historical perspective, used I to represent the French phrase “intensité de courant,” or current intensity.) In honor of Ampère, the unit of current is the Ampere, or Amp, or A, which is one Coulomb per second.

Current is defined as the rate at which charge passes through a specified surface area.

(Equation 10.5)

Often, that surface area will be the cross-section of a conductor, as shown in Figure 10.2.

I Figure 10.2. Current is defined as the rate at which charge passes through a surface.

In this course, we consider the fact that the current can vary from point to point, just as charge density can vary from point to point. Therefore, just as we use v to represent charge density, we also use J to represent current density. The units of J are A/m2, and if the current density is constant across a surface, the current density is just the total current divided by the total surface area:

(Equation 10.6)

If the current density varies across the surface area being considered, then we can perform a surface integral to determine the total current.

(Equation 10.7)

Example 10.6: Given the following current density, calculate the current flowing through the surface shown.

z=2 dS

z=1 y x=1

x=2 y=1 y=2 x

Example 10.7: Determine the total current flowing through a cylindrical wire of radius 2mm if the current density varies with radius as follows. If is in units of meters, then J will be in units of A/m2.

10.3 Ohm’s Law

It turns out that based on empirical observation, many materials demonstrate a current that is proportional to the total voltage applied across the length of the material. This is known as Ohm’s Law, and its macroscopic (bulk) version is shown below:

(Equation 10.8)

If we then take the definition of current density (Equation 10.6) and substitute Equation 10.8 for I, multiply the numerator and denominator by L, and then rearrange the terms, we obtain:

(Equation 10.9)

Assuming that the electric field is constant inside the material, we can calculate E as follows:

(Equation 10.10)

And we define , the conductivity to be:

(Equation 10.11)

Substituting Equation 10.10 and 10.11 into Equation 10.9, we find:

(Equation 10.12)

This is the microscopic (point) form of Ohm’s Law, and it is one of the three constitutive equations that accompany Maxwell’s Equations. Thus, it is one of the seven most important equations in this class.

The conductivity, , is a fundamental characteristic of the material being used. The units are Siemens/meter or 1/(∙m). Conductivity varies hugely according to the material being used. It is very large for conductors and almost zero for insulators. The following table lists the approximate conductivity for several materials.

Table 10.1 Conductivity (∙m)-1 Material Conductivity Copper 107 Graphite 106 Sea Water 100 Silicon 10-2 Rock Salt 10-4 Distilled Water 10-6 Marble 10-8 Mica 10-10 Paraffin 10-12 Fused Quartz 10-14

Example 10.8: A cylindrical resistor made of undoped silicon is 2cm in radius and 10 cm long. A voltage of 10V is applied across the length of the resistor. Calculate the following quantities related to this resistor: (a) Resistance, (b) Current, (c) Current density, and (d) Electric field. Assume the current density is uniform across the cross-section of the resistor.

10.4 Power Density

When current flows through a material, power is dissipated. The amount of power dissipated depends on the electric field and the current density. Since voltage is a measure of how much energy is required to move a charge across the material, and current is a measure of how quickly

9 charges are moving, the product of voltage and current yields the energy used per unit time, which is power. This is a universal result that is true for any type of circuit element.

(Equation 10.13)

Equation 10.13 is true for the entire device on a large scale, but we are also interested in microscopic variations in power dissipation. As with charge density and current density, we will consider power dissipation at a microscopic level, we will also consider power density, which is represented by a lower-case p. For a uniform material with uniform current density, we find that power density can be calculated as follows:

(Equation 10.14)

We can generalize this result to a vector representation, which allows for microscopic variations in current density and electric field.

(Equation 10.15)

Such variations might be caused by microscopic variations in the material conductivity, or it could be caused by the geometry of the problem.

If we want to know the total power dissipated within a three-dimensional region of current density and electric field, we can take a closed volume integral of the power density. (A volume integral is just like a line integral or a surface integral, except it is over a three-dimensional region of space. As a result, it will require triple integration.)

(Equation 10.16)

Example 10.9: What is the total power dissipated by the resistor described in Example 10.8?

10.5 Summary

 Line integrals are taken over a specified one-dimensional path between two points. First, you calculate the dot product between the vector field and a vector that is tangential to each point along the path, and then you integrate the result over the path.  Closed-loop integrals have the same beginning and ending point, and the result of the integration is independent of where it is started and ended.  Conservative fields will have closed-loop integral of zero for all possible paths. Non- conservative fields do not.  Continuous electric current is caused by charges always trying to go to their lowest potential energy state, but if an external force is applied to the system, this can lead to continuous motion of charges.  Electric current is defined as the rate at which charge passes through a specified surface area.

 The current density, J, is the current divided by the surface area, and current can be found by integrating current density over an area.

 Ohm’s Law can be written on a macroscopic level or in point form:

 Power density can also be calculated for the entire device or in point form: