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Electrodynamics Learning Material
Tutorial ID	Title	Tutorial	Video Tutorial	Revision Notes	Revision Questions
15.1	Electric Current. Current Density

In this Physics tutorial, you will learn:

Definition of electric current
Why electric current is that important in daily life?
What is charge density? How many types of charge density are there?
What is the difference between charge density and current density?
What is direct current?
What direction does the current flow in a circuit?
How can we find the electric field using the help of charge density and Gauss Law?

Introduction

Does static electricity (stationary charges) do any work? Is static electricity indispensable for the society in today's world? Why?

What kind of electricity do we use for most applications in daily life?

What is the direction of charges flow through a conductor?

In this tutorial, we will discuss about electric current, which is one of prerequisites for the existence of electricity. You can also check the answer of the above questions after reading the content and the solved examples.

Moving Charges. The Meaning of Electric Current

Stationary charges do not do any work. In fact, there are some useful applications of static electricity in daily life such as electrostatic precipitator, paint spray, photocopying machine, etc. as explained in the previous chapter; however, they are not really indispensable for the today's life. Most applications related to electricity in today's world use moving charges (dynamic electricity or electrodynamics) to operate. With moving charges, we usually have in mind electrons, as they are able to move freely between the atoms of conductor. For this reason, electrons are widely recognized as the main carriers of electricity. As we have explained earlier, protons (positive charges) are locked inside the nuclei of conductors and therefore they are not able to move. However, we will see later that in some specific cases, positive charged ions are also carriers of electricity, so we can say that not only negative charges are able to flow but this is an ability of positive charges as well.

Electric charges (typically electrons) always flow from places where there are more electrons to the places in which there are less electrons. This behaviour is similar to the behaviour of water when two containers with different levels of water are connected through a pipe. Water starts flowing from the higher level to the lower level container. This flow stops only after a balance in water level between the two containers is reached.

Physics Tutorials: This image provides visual information for the physics tutorial Electric Current. Current Density

The same thing occurs to electric charges as well. If there is some misbalance (a potential difference) between the numbers of free charges in the two extremities of a conductor, a charge flow takes place until both extremities have the same density of free charges.

By definition, electric current, is the amount of electric charges flowing through any point of conductor in the unit time.

The symbol of electric current in formulae is I. This is because the term "electric current" is an abbreviation of the longer term "intensity of electric current", but we use the short term "electric current" or simply "current" in conversational form.

Mathematically, we have

I = Q/t

The unit of electric current, Ampere (amp), A, is one of the seven fundamental units in the SI system used in science. Rearranging the above formula, we obtain the unit of electric charge, Coulomb in fundamental SI terms. Thus, since

Q = I ∙ t

it is easy to see that Coulomb = Ampere ∙ second in the SI system of units.

For example, if a device indicates the value 0.2 A on the display plate, it means there are 0.2 A ∙ 1 s = 0.2 C of charge flowing through the device in every second.

Charge Density and Current Density. Why are they Different?

Let's consider again the example with the two containers discussed in the previous paragraph. We assume the density of flowing material (water) and that of the surrounding environment as equal in both containers when conclude that the flow will stop when water reaches the same level in both containers. However, if the container on the right has a higher air density than the container on the left, the equilibrium is settled when water is not at the same level in both containers (remember the effects of capillary action and air pressure explained earlier on the liquid flow).

Another example: if containers are not identical, the water flow stops when water reaches the same level in both container, regardless the amount of water they hold.

The same phenomenon occurs to electric charges as well. It is not the number of free charges in both ends of a conductor the main factor that determines the electricity flow but it is the current density instead. For example, when we connect together two conducting spheres of different sizes through a conducting wire, the charge flow stops when both spheres reach the same charge density, not the same number of extra charges, as it is obvious the larger sphere will have more extra charges than the smaller sphere when the equilibrium is established.

In the above paragraph there were two concepts mentioned: charge density and current density. They do not represent the same thing. Let's explain their meaning providing some examples as an illustration.

a. Charge Density

Charge density shows how close electric charges are to each other in a conductor. There are three variants of charge density.

Linear charge density, λ. It shows how close the charges are to each other in a long and very thin conducting wire of length L. Linear charge density is calculated by the equation
λ = Q/L
and it is measured by Coulombs per metre [C/m].
Surface charge density, σ. It shows how close the charges are to each other in a surface of area A. Surface charge density is calculated by the equation
σ = Q/A
and it is measured by Coulombs per square metre [C/m2].
Volume charge density, ρ. It shows how close the charges are to each other in a space of volume V. Volume charge density is calculated by the equation
ρ = Q/V
and it is measured by Coulombs per cubic metre [C/m3].

Example 1

What is the linear charge density along the height, surface charge density and volume charge density if 24 μC of charge are distributed uniformly on the cylinder shown in the figure?

Solution 1

From the figure, we can see that:

L = 10 cm = 0.1 m = 10-1 m
R = 4 cm = 0.04 m = 4 × 10-2 m

Also, we have Q = 24 μC = 2.4 × 10-5 C.

Thus, for the linear charge density, we obtain

λ = Q/L
= 2.4 × 10^-5 C/10^-1 m
= 2.4 × 10^-4 C/m

Since the total area of cylinder is the sum of the two circular bases and the lateral face (which is a rectangle whose length is equal to base circumference and width is equal to the height of cylinder), we obtain for the surface charge density

σ = Q/A
= Q/2πR² + 2πR ∙ h
= Q/2πR ∙ (R + h)
= 2.4 × 10^-5 C/2 ∙ 3.14 ∙ 0.04 m ∙ (0.04 m + 0.1 m)
= 2.4 × 10^-5 C/3.5 × 10^-2 m²
= 0.686 × 10^-3 C/m²
= 6.86 × 10^-4 C/m²

As for the volume charge density, we have to consider the volume of cylinder, which is

V = πR² ∙ h
= 3.14 ∙ (4 × 10^-2 m)² ∙ 10^-1 m
= 5.024 × 10^-4 m³

Therefore, we obtain for the volume charge density

ρ=Q/V
= 2.4 × 10^-5 C/5.024 × 10^-4 m³
= 4.78 × 10^-2 C/m³

Remark! From all results found above, it is more realistic to discuss about area charge density than for the other two types of charge density, as it is a known fact that electric charges are distributed throughout the outer surface of a conductor. Only when the amount of charge is too large, free electrons are found inside the inner volume of conductor as well. On the other hand, if the conducting wire is as thin as to allow only one electron flowing through it, then we can discuss about linear charge density.

b. Current density

Charge density is a quantity that regards situations when there is a kind of equilibrium, i.e. when the charges do not move after being distributed throughout the conductor. It is a known fact that in universe everything is dynamic (in motion) though. Of course, the static approach is very useful to explain the related concepts in an easier way; however, it is the dynamic approach the one that gives us a clearer picture on how the world of electricity works.

For this reason, we introduce the concept of current density, J, which is an extension of that of charge density. Current density is a vector quantity that gives the rate of current flow through a certain area A. Mathematically, the current density is calculated by

J = I/A

where I is the current and A is the area in which the current flows (usually the cross-sectional area of conductor), similar to the area discussed in the tutorial that describes the electric flux explained in the previous chapter.

However, in vector form we use another equation to describe the current density. It is

J⃗ = ρ ∙ v⃗

where ρ is the volume charge density and v is known as the "drift velocity", i.e. the net velocity of charges movement in a certain direction (usually in the direction determined by the push of the electric source along the conducting wire). This is because the charges movement is quite complex due to the continuous collision with other particles; however, we are interested only in the direction in which they can do work, i.e. in the direction in which electricity moves. Drift motion is similar in concept to thermal motion. There is one fundamental difference between these two concepts however. Drift motion gives a resultant in a single direction for all charges (the direction of electrons flow along the conducting wire), while the resultant direction of thermal motion is zero because such a motion is random. (Another difference is that thermal motion regards atoms and molecules of substance while drift motion regards to electric charges only).

Let's prove the equivalence between the two above formulae for current density through the method of dimensional analysis (by considering the units). Thus, when considering the first (scalar) formula, we obtain

[Unit of current density J] = [Unit of electric current I]/[Unit of area A]
= [A]/[m² ]
= [A/m² ]

while when considering the second (vector) formula, we obtain

[Unit of current density J] = [Unit of volume charge density ρ] ∙ [Unit of drift velocity v]
= [C/m³ ] ∙ [m/s]
= [C/(s ∙ m² )]
= [C/s] ∙ [1/m² ]
= [A] ∙ [1/m² ]
= [A/m² ]

As you see, the units in both cases are equal, so both formulae are true for the current density.

Example 2

Find the drift velocity of an electron moving in a 10 m long copper wire connected to a 12 V battery at room temperature if the mean free time between two collisions is τ = 3 × 10-14 s? Take the charge of electron as e = 1.6 × 10-19 C and its mass is m_e = 9.1 × 10-31 kg.

Solution 2

The electron accelerates uniformly from rest until it reaches the maximum velocity. This means the average drift velocity is half of the maximum drift velocity, similar to when an object starts moving from rest at constant acceleration in kinematics. Then it stops after hitting another particle, starts accelerating again and so on. This process repeats itself periodically at time intervals τ. Therefore, we can write

v_drift = 1/2 ∙ ∆v
= 1/2 ∙ a ∙ τ
= 1/2 ∙ F_e/m_e ∙ τ
= 1/2 ∙ e ∙ E/m_e ∙ τ
= 1/2 ∙ e ∙ ∆V/m_e ∙ d ∙ τ
= e ∙ ∆V ∙ τ/2m_e ∙ d
= (1.6 × 10^(-19) C) ∙ (12 V) ∙ (3 × 10^(-14) s)/2 ∙ (9.1 × 10^(-31) kg) ∙ (10 m)
= 3.16 × 10^-3 m/s
= 3.16 mm/s

This value is very small compared to the velocity of electric current, which is close to the speed of light, i.e. at the order of 105 km/s. This means the effect of electric current is mostly due to its wave effect than due to the charges movement.

Direct Current. The Direction of Current Flow

Direct current (DC) is the simplest type of current. The main producers of direct current are batteries, whose positive and negative terminals are well defined. This means the current has a single direction of flow throughout an electrical circuit. In the early days of electricity discovery, it was thought that positive charges move from the positive to negative terminal of battery. However, with the discovery of electron, it was made clear that there are electrons the particles that flow through the conductor, not the positive charges (protons). However, by agreement between scientists, it was decided to keep the actual direction of motion (from positive to negative). This direction was called the "conventional direction of current flow" and it is opposite to the direction of electrons flow (which is from negative to positive terminal of battery). Let's consider an analogy to clarify this point.

Suppose there are four people and five chairs available in a certain column of a theatre hall. Initially the firs chair is free and the four people are sitting from the second to the fifth chair. When the person who is sitting on the second chair realizes that nobody is coming to sit on the first chair, stands up and moves there. Similarly, the person who is sitting to the third chair moves to the second chair that is left free from the previous person, and so on. This process continues until the last chair remains free, because all the four people present now occupy the first four chairs of the column.

Thus, despite the true movement is the shift of a person by one position due left, it seems like an empty chair is shifting by one position due right. The stationary empty chairs here symbolize the positive charges (protons) while the moving people symbolize the electrons. Now, it is clear why we consider the direction of current from positive to negative despite electrons move in the opposite direction. After all, the result of calculations does not change.

Appendix - Charge Density, Electric Field and Gauss Law

Now that we explained the meaning of charge density in all possible forms (linear, surface and volume charge density), let's recall Gauss Law and extend its explanation further by taking into consideration the new concepts. Remember that Gauss Law involves electric flux Φ and its general mathematical form is

Φ = Q/ϵ₀

while the formula derived directly from definition of electric flux is

Φ = E ∙ A ∙ cosθ

where E is the electric field, A is the area vector and θ is the angle between the area vector and electric field vector.

a. Transforming Gauss Law using the surface charge density concept

Let's consider a charged conductor of surface area A. Its surface charge density obviously is σ = Q/A where Q is the charge. From Gauss Law, we know that

Φ = Q/ϵ₀ =E ∙ A ∙ cos⁡θ

For simplicity, let's consider the field lines as parallel to the area vector, i.e. cos θ = 1 because θ = 0⁰.

Given that Q = σ ∙ A, we can write

σ ∙ A/ϵ₀ = E ∙ A

Simplifying the area A from both sides, we obtain for the electric field in terms of surface charge density

E = σ/ϵ₀

The above formula means that the magnitude of electric field outside a conductor is proportional to the surface charge density on the conductor. This is one of the most important assertions in Electromagnetism.

b. Transforming Gauss Law using the linear charge density concept. Cylindrical symmetry

Let's consider a long and very thin bar carrying a uniform linear charge density, λ. We want to find an expression for the electric field E in terms of linear charge density and distance r from the bar. For this, we must consider a cylinder of radius r and height h, and then, find the electric field produced on the lateral surface of it.

Again, the area vector A is parallel to the direction of electric field E. This means cos θ = 1.

From the definition of electric flux, we have

Φ = E ∙ A ∙ cos⁡θ
= E ∙ A ∙ 1
= E ∙ A
= E ∙ 2πr ∙ h

From Gauss Law, we have

Φ = Q/ϵ₀
= λ ∙ h/ϵ₀

Therefore, combining the two above equations, we obtain

E ∙ 2πr ∙ h = λ ∙ h/ϵ₀

Simplifying h from both sides and rearranging in order to isolate E, we obtain

E = λ/2π ∙ r ∙ ϵ₀

Example 3

What is the electric field produced by a long bar of linear charge density λ = 600 μC/cm at a distance of 4 m from the bar?

Solution 3

Clues:

λ = 600 μC/cm = 60000 μC/m = 0.06 C/m = 6 × 10-2 C/m
r = 4 m
ϵ0 = 8.85 × 10-12 F/m
E = ?

From the equation of electric field in terms of linear charge density, we have

E = λ/2π ∙ r ∙ ϵ₀
= 6 × 10^-2 C/m/2 ∙ 3.14 ∙ (4 m) ∙ (8.85 × 10^(-12) F/m)
= 2.7 × 10⁸ V/m

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