Electric Current

Electric current is the rate of flow of electric charge. It is a fundamental concept in the study of electricity and is crucial for understanding how electrical devices work. Imagine a river flowing; the water molecules moving from one place to another constitute a flow. Similarly, in a conductor, it is the charged particles, usually electrons, that move, creating an electric current.

The direction of conventional current is defined as the direction of flow of positive charge. Even though in most metallic conductors, it is the negatively charged electrons that actually move, the convention remains. This convention was established by Benjamin Franklin.

Quantifying Electric Current

Mathematically, electric current ($I$) is defined as the amount of charge ($Q$) passing through any cross-sectional area of a conductor per unit time ($t$).

The formula is: $I = \frac{Q}{t}$

Here, $Q$ is the total charge that has passed and $t$ is the time taken for this charge to pass.

The SI unit of electric current is the Ampere (A), named after the French physicist André-Marie Ampère. One Ampere is defined as the flow of one Coulomb of charge per second.

$1 \text{ A} = 1 \frac{\text{C}}{\text{s}}$

If the rate of flow of charge varies with time, we use the instantaneous current formula, which is the derivative of charge with respect to time:

$I = \frac{dQ}{dt}$

This means that if we know the charge as a function of time, $Q(t)$, we can find the current at any instant by differentiating $Q(t)$ with respect to $t$.

Conversely, if we know the current as a function of time, $I(t)$, the total charge $Q$ that flows through a cross-section in a time interval from $t_1$ to $t_2$ can be found by integrating the current over that time interval:

$Q = \int_{t_1}^{t_2} I(t) dt$

Example Scenario

Consider a wire through which 5 Coulombs of charge flow in 10 seconds. The electric current in the wire would be:

$I = \frac{5 \text{ C}}{10 \text{ s}} = 0.5 \text{ A}$

If the charge flowing through a point in a circuit is given by $Q(t) = (2t^2 + 3t + 1)$ Coulombs, where $t$ is in seconds, then the instantaneous current at $t=2$ seconds is:

$I(t) = \frac{dQ}{dt} = \frac{d}{dt}(2t^2 + 3t + 1) = 4t + 3$

At $t=2$ seconds, $I(2) = 4(2) + 3 = 8 + 3 = 11$ Amperes.

Types of Current

There are two main types of electric current:

• Direct Current (DC): In direct current, the charge flows consistently in one direction. Batteries and solar cells provide DC.
• Alternating Current (AC): In alternating current, the direction of charge flow reverses periodically. The electricity supplied to our homes is typically AC.

Understanding the direction and magnitude of current is essential for analyzing electrical circuits and for safety precautions.

Drift Velocity

When a conductor, like a copper wire, is not connected to any voltage source, the free electrons within it are in constant, random motion. They collide with the positive ions of the metal lattice and with each other. Their motion is chaotic, and there is no net movement of charge in any particular direction. Therefore, no current flows.

However, when an electric field is applied across the conductor (by connecting it to a battery, for example), these free electrons experience a force in the direction opposite to the electric field (since electrons are negatively charged). This force causes them to accelerate.

But their motion is not unimpeded. They still collide with the ions in the lattice. These collisions cause them to lose their accelerated velocity and scatter. After each collision, they are again accelerated by the electric field, only to collide again.

As a result of this continuous acceleration and frequent collisions, the electrons do not accelerate indefinitely. Instead, they acquire a small, average velocity in the direction opposite to the electric field. This average velocity is called the **drift velocity** ($v_d$).

Drift velocity is a very slow, directed motion superimposed on the much faster random thermal motion of the electrons. It is this drift velocity that is responsible for the electric current in the conductor.

Factors Affecting Drift Velocity

The drift velocity depends on several factors:

• Electric Field Strength ($E$): A stronger electric field exerts a greater force on the electrons, leading to a higher drift velocity.
• Nature of the Conductor: Different materials have different lattice structures and electron densities, affecting how easily electrons can move.
• Temperature: Higher temperatures lead to more vigorous vibrations of the lattice ions, resulting in more frequent collisions and thus a lower drift velocity.

Drift Velocity and Collisions

Let's consider an electron with charge $e$ and mass $m$. When an electric field $E$ is applied, the force on the electron is $F = -eE$. This force causes an acceleration $a = \frac{F}{m} = -\frac{eE}{m}$.

Due to collisions, the electrons do not maintain this acceleration indefinitely. Let $\tau$ be the average time between successive collisions of an electron with the ions of the conductor. This time $\tau$ is called the relaxation time or mean free time.

An electron starts with some random velocity and is accelerated for time $\tau$. Its velocity just before the next collision, due to the electric field, would be $a\tau = -\frac{eE}{m}\tau$. Since the initial velocity is random and the collisions are frequent, the average velocity gained by the electrons in the direction of the field is the drift velocity.

Therefore, the drift velocity ($v_d$) is given by:

$v_d = a \tau = -\frac{eE}{m}\tau$

The negative sign indicates that the drift velocity is in the opposite direction to the electric field, which is consistent with electrons (negative charges) drifting against the field.

The magnitude of the drift velocity is:

$|v_d| = v_d = \frac{eE\tau}{m}$

This equation shows that drift velocity is directly proportional to the applied electric field strength and the relaxation time.

Mobility

Mobility is a measure of how easily charge carriers can move through a material under the influence of an electric field. It is defined as the magnitude of the drift velocity acquired by the charge carriers per unit electric field.

The symbol for mobility is $\mu$ (mu).

Mathematically, mobility is expressed as:

$\mu = \frac{|v_d|}{E}$

Using the expression for drift velocity, we can substitute $|v_d| = \frac{eE\tau}{m}$:

$\mu = \frac{\frac{eE\tau}{m}}{E}$

$\mu = \frac{e\tau}{m}$

This formula shows that the mobility of charge carriers depends on the average time between collisions ($\tau$) and their mass ($m$). It is independent of the electric field strength.

Units of Mobility

The SI unit of mobility can be derived from its definition. Since $v_d$ is in m/s and $E$ is in V/m (Volts per meter), the unit of $\mu$ is:

$\frac{\text{m/s}}{\text{V/m}} = \frac{\text{m}^2}{\text{V} \cdot \text{s}}$

So, the SI unit for mobility is square meters per Volt-second ($m^2 V^{-1} s^{-1}$).

Significance of Mobility

Mobility is an important property of a material because it directly influences its electrical conductivity. Materials with higher mobility allow charge carriers to move more freely, resulting in higher conductivity.

For different charge carriers, the mobility values can vary significantly. For example, electrons generally have higher mobility than holes in semiconductors because they are lighter and experience fewer scattering events.

In metals, the charge carriers are electrons, and their mobility is relatively high. In semiconductors, both electrons and holes can act as charge carriers, and their mobilities are generally lower than in metals but can be varied by doping.

For example, in silicon, the electron mobility is around $1400 \text{ cm}^2 V^{-1} s^{-1}$ and the hole mobility is around $450 \text{ cm}^2 V^{-1} s^{-1}$. These values are crucial for designing transistors and other semiconductor devices.

Relation Between Current and Drift Velocity

We can establish a direct relationship between the electric current ($I$) flowing through a conductor and the drift velocity ($v_d$) of the charge carriers.

Consider a segment of a conductor of length $L$ and uniform cross-sectional area $A$. Let $n$ be the number of free charge carriers (e.g., electrons) per unit volume in the conductor. The total number of free charge carriers in this segment of length $L$ is $N = nAL$.

If each charge carrier has a charge $q$ (for electrons, $q = -e$), then the total charge ($Q$) in this segment is $Q = Nq = (nAL)q$.

Now, imagine an electric field applied across the conductor. The charge carriers drift with an average velocity $v_d$. In time $\Delta t$, the charge carriers move a distance $\Delta x = v_d \Delta t$.

Consider the charge that flows across the end face of the conductor in time $\Delta t$. This charge consists of all the charge carriers that were within a distance $v_d \Delta t$ from the end face. The volume containing this charge is $A(v_d \Delta t)$.

The total charge $Q$ that passes through the cross-sectional area $A$ in time $\Delta t$ is:

$Q = (\text{number density of charge carriers}) \times (\text{volume containing the charge}) \times (\text{charge per carrier})$

$Q = n \times (A \cdot v_d \Delta t) \times q$

The current $I$ is the rate of flow of charge, so $I = \frac{Q}{\Delta t}$.

$I = \frac{n A v_d \Delta t q}{\Delta t}$

$I = n A v_d q$

This is a very important equation relating current to drift velocity.

For Metallic Conductors (Electrons)

In metallic conductors, the charge carriers are electrons, so $q = -e$. The number density of free electrons is $n$. The current is given by:

$I = n A v_d (-e)$

$I = -n A v_d e$

The negative sign indicates that the direction of current (conventionally defined by positive charge flow) is opposite to the direction of electron drift. If we consider the magnitude of the current and drift velocity:

$|I| = n A |v_d| e$

Or, simply using $v_d$ as the magnitude of drift velocity:

$I = n A v_d e$

This equation clearly shows that the current is directly proportional to the number density of free electrons ($n$), the cross-sectional area ($A$), and the drift velocity ($v_d$).

Connecting with Mobility

We know that $v_d = \mu E$, where $\mu$ is the mobility and $E$ is the electric field. Substituting this into the current equation:

$I = n A (\mu E) q$

For electrons, $q = -e$, so the magnitude of current is $I = n A \mu E e$.

Example Calculation

A copper wire has a cross-sectional area of $3 \times 10^{-6} \text{ m}^2$ and a free electron density of $8 \times 10^{28} \text{ m}^{-3}$. If a current of $5.4$ A flows through the wire, calculate the drift velocity of the electrons. (Take $e = 1.6 \times 10^{-19}$ C).

We use the formula $I = n A v_d e$.

Rearranging for $v_d$:

$v_d = \frac{I}{n A e}$

Plugging in the values:

$v_d = \frac{5.4 \text{ A}}{(8 \times 10^{28} \text{ m}^{-3}) \times (3 \times 10^{-6} \text{ m}^2) \times (1.6 \times 10^{-19} \text{ C})}$

$v_d = \frac{5.4}{8 \times 3 \times 1.6 \times 10^{(28 - 6 - 19)}} \text{ m/s}$

$v_d = \frac{5.4}{38.4 \times 10^3} \text{ m/s}$

$v_d \approx 0.14 \times 10^{-3} \text{ m/s}$

$v_d \approx 1.4 \times 10^{-4} \text{ m/s}$

This demonstrates that drift velocities are typically very small, on the order of millimeters per second.

Ohm's Law

Ohm's Law is one of the most fundamental laws in electricity, describing the relationship between voltage, current, and resistance in an electrical circuit. It was formulated by the German physicist Georg Simon Ohm in 1827.

Ohm's Law states that for a metallic conductor at a constant temperature, the electric current flowing through it is directly proportional to the potential difference (voltage) applied across its ends.

In simpler terms, if you increase the voltage across a resistor, the current through it increases proportionally, provided the resistance remains constant.

Mathematical Formulation

Mathematically, Ohm's Law is expressed as:

$V \propto I$ (at constant temperature)

This proportionality can be turned into an equation by introducing a constant of proportionality, which is the resistance ($R$) of the conductor.

$V = IR$

Where:

• $V$ is the potential difference across the conductor (measured in Volts, V).
• $I$ is the electric current flowing through the conductor (measured in Amperes, A).
• $R$ is the resistance of the conductor (measured in Ohms, $\Omega$).

This equation can be rearranged to find current ($I = V/R$) or resistance ($R = V/I$).

The resistance ($R$) is a property of the conductor that opposes the flow of current. It depends on the material of the conductor, its length, and its cross-sectional area, as well as its temperature.

Conditions for Ohm's Law

Ohm's Law is not universally applicable to all materials and conditions. It holds true for:

• Ohmic Materials: Materials that obey Ohm's Law are called ohmic conductors. Most metallic conductors (like copper, aluminum, silver) behave ohmically at constant temperatures.
• Constant Temperature: The law is strictly valid only when the temperature of the conductor remains constant. When current flows through a conductor, it heats up due to collisions of charge carriers with the lattice. This increase in temperature can change the resistance of the material, causing it to deviate from Ohm's Law.

Materials or devices that do not obey Ohm's Law are called non-ohmic. Examples include diodes, transistors, and light bulbs (where the filament's resistance changes significantly with temperature).

Microscopic View of Ohm's Law

We can derive Ohm's Law from the concept of drift velocity. We have the current $I = n A v_d q$.

We also know that $v_d = \frac{eE\tau}{m}$ (for electrons, with $q=-e$). So, $v_d = \frac{qE\tau}{m}$ for a general charge carrier $q$.

Substituting this into the current equation:

$I = n A \left(\frac{qE\tau}{m}\right) q = \frac{n A q^2 \tau E}{m}$

Now, consider the conductor of length $L$ and cross-sectional area $A$. The potential difference $V$ across it is related to the electric field $E$ by $V = EL$. So, $E = V/L$.

Substituting $E = V/L$:

$I = \frac{n A q^2 \tau}{m} \left(\frac{V}{L}\right)$

Rearranging this equation to get $V$ on one side:

$V = \left(\frac{mL}{n A q^2 \tau}\right) I$

Comparing this with Ohm's Law $V = IR$, we can identify the resistance $R$ as:

$R = \frac{mL}{n A q^2 \tau}$

This expression for resistance shows that $R$ is constant for a given conductor at a constant temperature (since $n, m, q, \tau, L, A$ are constant). This provides a microscopic justification for Ohm's Law.

Conductivity and Resistivity

The reciprocal of resistance is conductance ($G = 1/R$). The reciprocal of resistivity is conductivity ($\sigma$).

From $R = \frac{mL}{n A q^2 \tau}$, we can write $R = \left(\frac{m}{n q^2 \tau}\right) \frac{L}{A}$.

The term $\rho = \frac{m}{n q^2 \tau}$ is the resistivity of the material.

So, $R = \rho \frac{L}{A}$. This is the macroscopic formula for resistance.

The conductivity $\sigma$ is the reciprocal of resistivity:

$\sigma = \frac{1}{\rho} = \frac{n q^2 \tau}{m}$

Using conductivity, the relationship $I = n A q v_d$ can be written as $I = n A q (\mu E) = n A q (\frac{\sigma}{nq}) E = \sigma A E$.

Since $I = \sigma A E$ and $V = EL$, we have $I = \sigma A (V/L)$, which leads to $V = \frac{1}{\sigma} \frac{L}{A} I$. Here, $R = \frac{L}{\sigma A} = \rho \frac{L}{A}$.

Ohm's Law in Different Forms

Ohm's Law can be expressed in several ways:

• Circuit Form: $V = IR$ (Relates voltage, current, and resistance in a circuit).
• Material Form: $\vec{J} = \sigma \vec{E}$ (Relates current density $\vec{J}$ to electric field $\vec{E}$ through conductivity $\sigma$). This is the most general form.

Example Application

A resistor has a resistance of $100 \Omega$. If a voltage of $5$ V is applied across it, what is the current flowing through it?

Using Ohm's Law, $I = \frac{V}{R}$.

$I = \frac{5 \text{ V}}{100 \Omega} = 0.05 \text{ A}$

This is equal to 50 milliamperes (mA).

If the voltage is increased to $10$ V, the current becomes $I = \frac{10 \text{ V}}{100 \Omega} = 0.1 \text{ A}$, demonstrating the direct proportionality.