Current Electricity Lecture Notes

3.1 INTRODUCTION

In Chapter 1, we considered all charges to be at rest, whether free or bound. However, charges in motion constitute an electric current. These currents can occur naturally in various situations. An example of such a phenomenon is lightning, in which charges flow from the clouds to the earth through the atmosphere, sometimes leading to disastrous consequences. The flow of charges in lightning is not steady; notwithstanding, we often encounter devices in our everyday life, like a torch or a cell-driven clock, where charges flow steadily, resembling water flowing smoothly in a river. In this chapter, we shall explore the fundamental laws concerning steady electric currents.

3.2 ELECTRIC CURRENT

Imagine a small area oriented perpendicular to the direction of charge flow. Both positive and negative charges can flow forward and backward across this area. Let:

• $q_+$ be the net amount of positive charge flowing in the forward direction across the area during a given time interval $t$.
• $q_-$ be the net amount of negative charge flowing in the forward direction across the area during the same time interval.
  Thus, the net amount of charge flowing across the area in the forward direction is given by:
  $q = q_+ - q_-$. This quantity is proportional to $t$ for a steady current.
  The electric current $I$ across the area in the forward direction is defined as:
  $I = \frac{q}{t} ag{3.1}$
  Note that if this value turns out to be negative, it implies a current in the backward direction. Generally, currents are not steady, hence we can define the current more generally. Let $riangle Q$ be the net charge flowing across a conductor's cross-section during the time interval $riangle t$ (i.e., between times $t$ and $(t + riangle t)$). Then, the current at time $t$ across this cross-section is defined as:
  $I(t) = \frac{ riangle Q}{ riangle t} \bigg|_{ riangle t o 0} ag{3.2}$
  In SI units, the unit of current is the ampere (A). An ampere is defined through the magnetic effects of currents, which will be discussed in the next chapter. Typically, an ampere is the order of magnitude of currents found in domestic appliances. For comparison, an average lightning carries currents on the order of tens of thousands of amperes, while currents in human nerves are on the scale of microamperes.

3.3 ELECTRIC CURRENTS IN CONDUCTORS

An electric charge experiences a force when an electric field is applied. If it's free to move, it contributes to the current. Free charged particles exist in nature, particularly in the upper atmosphere (the ionosphere). However, within atoms and molecules, the negatively charged electrons and positively charged nuclei are bound together, inhibiting free movement. For instance, a gram of water comprises approximately $10^{22}$ molecules, which are closely packed. In some materials, electrons remain bound and will not accelerate under the influence of an electric field.
Conversely, in metals, some electrons can move freely within the material, enabling the conduction of electric currents when an electric field is applied. The discussion will primarily focus on solid conductors, where currents arise from negatively charged electrons moving against a backdrop of fixed positive ions.

When no electric field is present, the electrons behave randomly due to thermal motion, colliding with fixed ions. Collisions yield unchanged speeds but randomize direction, such that there is no preferential direction for electron velocities—resulting in no net electric current. Upon applying an electric field, consider a cylindrical conductor of radius $R$. Imagine placing two thin circular dielectric discs with positive charge $+Q$ on one disc and negative charge $-Q$ on the other. An electric field is created that directs from positive to negative charge. The electrons will be accelerated toward the positive charge $+Q$, generating an electric current as they move to neutralize the charges. However, this current will cease once the charges are neutralized, unless we replenish the charges, a mechanism we will discuss later using cells or batteries.

3.4 OHM’S LAW

In 1828, G.S. Ohm identified a fundamental law regarding the flow of currents, predating the understanding of the physical mechanisms driving such flows. If a current $I$ flows through a conductor with a potential difference $V$ between its ends, Ohm's law states that:
$V = R imes I ag{3.3}$
Here, $R$ denotes the resistance of the conductor. The unit of resistance in SI is the ohm (Ω). This resistance is influenced not only by the material of the conductor but also by its dimensions.
To derive how $R$ depends on dimensions, imagine a conductor in the form of a slab with length $l$ and cross-sectional area $A$. If we place two such slabs side by side, making the total length $2l$, the same current $I$ flows through both while the potential difference across the combination is $2V$. We find the resistance of this combination $R_C$ as follows:
$R_C = \frac{V}{I} = 2 \frac{V}{I} = 2R ag{3.4}$
Thus, doubling the length of a conductor doubles the resistance, leading to:
$R o l ag{3.5}$
Next, if we cut the slab lengthwise, this results in two identical halves, each with a length $l$ and cross-sectional area $\frac{A}{2}$. While the full slab has resistance $R$, each half has resistance:
$R_1 = 2 \frac{V}{I} = 2R ag{3.6}$
Consequently, halving the cross-sectional area doubles the resistance, leading to:
$R o \frac{1}{A} ag{3.7}$
By combining these two relationships, we have:
$R o \frac{l}{A} ag{3.8}$
This equation indicates that for a given conductor, its resistance $R$ relates to resistivity
ho as follows:
R =
ho rac{l}{A} ag{3.9}
The constant of proportionality
ho, known as resistivity, depends on the material of the conductor but not its dimensions. Thus, Ohm’s law can be expressed in terms of resistivity as:
I = rac{V imes A}{
ho imes l} ag{3.10}
The current per unit area (normal to the current), $\frac{I}{A}$, is referred to as current density denoted by $j$ with SI units of $A/m^2$. Additionally, if $E$ is the magnitude of the uniform electric field in a conductor of length $l$, we know the potential difference $V$ is represented as:
$V = E imes l$. Thus, we can write:
E = rac{j}{
ho} ag{3.11}
This relationship can be expressed in vector form, where:
extbf{j} =
ho extbf{E} ag{3.12}
Furthermore, we can rearrange to show:
extbf{j} = rac{1}{
ho} extbf{E} ag{3.13}
Here, rac{1}{
ho} is called conductivity. Ohm's law can thus also be stated in the form of Eq. (3.13) as an equivalently valid expression.

3.5 DRIFT OF ELECTRONS AND THE ORIGIN OF RESISTIVITY

It has been previously stated that electrons encounter collisions with heavy fixed ions; after these collisions, they retain their speed but change direction. Importantly, the average velocity of electrons equals zero due to their random directions. Let $N$ denote the total number of electrons, with the velocity of the $i^{th}$ electron represented as $v_i$. We calculate the average velocity at a given time as:
$V_{avg} = \frac{1}{N} extstyle \footnotesize{\bigg| \bigg| extit{زمید}\bigg| \bigg| }$

Consider the conditions under which an electric field is applied. When this occurs, electrons are accelerated according to the formula:
$a = -\frac{eE}{m} ag{3.15}$
Where:

• $-e$ represents the charge of the electron, and

• $m$ symbolizes the mass of the electron.
  For the $i^{th}$ electron at time $t$, if $t_i$ indicates the time elapsed since the last collision, then the velocity of the electron after experiencing this electric field can be expressed as:
  $v_i(t) = v_i + a imes t_i = v_i + \frac{-eE}{m} imes t_i ag{3.16}$
  This expression shows how the electron is accelerated over time, and we can further illustrate the drift of electrons in the presence of the electric field using a schematic image. An electron stemming from point A and traveling to point B will experience repeated collisions but will ultimately continue its movement toward B, thus resulting in a net drift opposite the electric field.

  Figure references for diagrams are cited but not included. They should demonstrate configurations and illustrations discussed, namely Figures 3.1, 3.2, and 3.3.