Basic Concepts and Ohm's Law

Electric Current And Circuit

The concept of electric current is central to the study of electricity. Fundamentally, electric current represents the flow of electric charge. Imagine a river where water particles are flowing; similarly, electric current is the flow of charged particles. In most common conductors like metals, these charged particles are negatively charged electrons. In other media, such as electrolytes (ionic solutions) or ionised gases (plasma), the charge carriers can be positive ions, negative ions, or both.

Quantitatively, electric current ($I$) is defined as the rate of flow of electric charge through a given cross-sectional area of a conductor. If a net charge $\Delta Q$ flows through a cross-section in time $\Delta t$, the average current is:

$ I_{avg} = \frac{\Delta Q}{\Delta t} $

For a current that may vary with time, the instantaneous current ($I$) at any moment is given by the differential form:

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

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Units of Electric Current

The SI unit of electric charge is the coulomb (C), and the SI unit of time is the second (s). Therefore, the SI unit of electric current is coulomb per second ($C/s$), which is called the ampere (A), in honour of André-Marie Ampère.

One ampere is defined as the flow of one coulomb of charge per second through any cross-section of a conductor. Smaller units like milliampere ($mA = 10^{-3} A$) and microampere ($\mu A = 10^{-6} A$) are also commonly used.

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Direction of Electric Current

Historically, before the discovery of electrons, it was assumed that electric current consisted of positive charges flowing. Therefore, the direction of electric current was defined as the direction of flow of positive charges. This is known as conventional current.

In metallic conductors, the charge carriers are electrons, which are negatively charged. Electrons flow from a region of lower potential (negative terminal of a battery) to a region of higher potential (positive terminal). The direction of electron flow is thus opposite to the direction of conventional current.

Unless otherwise specified, when we refer to electric current, we usually mean the conventional current, which flows from higher potential to lower potential.

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Electric Circuit

An electric circuit is a closed loop or pathway that allows electric charge to flow. A simple electric circuit typically consists of:

• A source of potential difference: Such as a battery or a power supply, which provides the electrical energy and drives the charge flow.
• Conductors: Usually wires made of materials with low resistance (like copper or aluminium), providing the path for current.
• A load or component: A device that consumes electrical energy and converts it into another form (e.g., a light bulb converts it into light and heat, a motor converts it into mechanical energy).
• A switch (optional but common): Used to open or close the circuit, controlling whether current flows. An open circuit breaks the path, stopping the current; a closed circuit completes the path, allowing current to flow.

For a current to flow continuously, the circuit must be closed and the potential difference must be maintained by the source.

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Types of Current

Electric currents can be broadly classified into two types:

• Direct Current (DC): Current that flows in only one direction. The magnitude of DC can be constant or varying, but the direction remains the same. Batteries are sources of DC.
• Alternating Current (AC): Current that periodically reverses direction. The magnitude also changes periodically. The power supplied to our homes is AC.

Electric Potential And Potential Difference

To understand why electric charge flows in a circuit, we need the concept of electric potential. Just as differences in gravitational potential cause mass to move, and differences in pressure cause fluid to flow, differences in electric potential cause electric charge to flow.

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Electric Potential

The electric potential ($V$) at a point in an electric field is defined as the amount of work done by an external force per unit positive test charge to move the charge from infinity to that point without any acceleration.

$ V = \frac{W}{q_0} $

Here, $W$ is the work done and $q_0$ is the positive test charge. Electric potential is a scalar quantity.

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Electric Potential Difference

More practically, it is the electric potential difference ($\Delta V$) between two points that drives the current in a circuit. The potential difference between two points A and B is defined as the work done by an external force per unit positive charge to move the charge from point A to point B without acceleration.

$ \Delta V = V_B - V_A = \frac{W_{AB}}{q_0} $

This work is done against the electric field forces. Alternatively, if the electric field does work $W_E$ in moving a charge $q_0$ from A to B, the potential difference is $\Delta V = V_B - V_A = -W_E/q_0$.

In a circuit, a source like a battery does work to move charges from its low potential terminal to its high potential terminal internally, thereby maintaining a potential difference across its terminals. This potential difference is what drives the current through the external circuit components.

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Units of Potential and Potential Difference

The SI unit of electric potential and potential difference is the volt (V), named after Alessandro Volta.

One volt is defined as the potential difference between two points if 1 joule (J) of work is done to move 1 coulomb (C) of positive charge from one point to the other.

$ 1 \text{ V} = \frac{1 \text{ J}}{1 \text{ C}} $

A device commonly used to provide and maintain a potential difference is an electric cell or a battery (a combination of cells).

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Measurement of Current and Potential Difference

Ammeter: Used to measure the electric current flowing through a component or a section of the circuit. It must be connected in series with the component so that the entire current passes through it. An ideal ammeter has zero resistance to avoid affecting the current being measured.

Voltmeter: Used to measure the potential difference across a component. It must be connected in parallel across the component so that it measures the potential difference between the two points it is connected to. An ideal voltmeter has infinite resistance so that it draws negligible current from the circuit, thereby not affecting the potential difference being measured.

Circuit Diagram

Communicating the layout and connections of an electric circuit is efficiently done using a circuit diagram. A circuit diagram is a graphical representation of an electrical circuit, where various components are depicted by standard symbols and lines represent connecting wires. This provides a clear and concise way to show how different components are connected in a circuit.

Using standardised symbols is crucial for universal understanding and interpretation of circuit designs, regardless of language or location.

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Standard Symbols for Electrical Components

Below is a table showing some common symbols used in circuit diagrams, widely accepted internationally and in India.

Component	Symbol	Description
Electric cell		A single unit that converts chemical energy into electrical energy, providing a potential difference. The longer line represents the positive terminal, and the shorter, thicker line represents the negative terminal.
Battery		A combination of two or more electric cells connected in series, increasing the total potential difference.
Switch (Open) or Plug key (Open)		A device used to break or open the circuit, preventing the flow of current.
Switch (Closed) or Plug key (Closed)		A device used to complete or close the circuit, allowing the flow of current.
Wire Joint		Indicates that two or more wires are electrically connected at a point.
Wires crossing (no joint)		Indicates that wires cross over each other physically but are not electrically connected.
Electric bulb or Lamp		A device that converts electrical energy into light and heat (represented as a resistor).
Resistor		A component designed to introduce a specific amount of resistance into a circuit, opposing current flow.
Variable resistance or Rheostat		A resistor whose resistance value can be adjusted. Used for controlling current or voltage.
Ammeter		An instrument used to measure the electric current flowing through a point in amperes. Connected in series.
Voltmeter		An instrument used to measure the electric potential difference across two points in volts. Connected in parallel.
Galvanometer		An instrument used to detect the presence and direction of small electric currents.

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Drawing a Simple Circuit Diagram

Let's represent a simple circuit consisting of a battery (made of two cells), a switch, a light bulb, an ammeter to measure the total current, and a voltmeter to measure the potential difference across the bulb.

We start by drawing the battery symbol. From one terminal (say, positive), draw a line to the switch symbol. From the other side of the switch, draw a line to the ammeter symbol (connected in series, so the line enters one terminal and exits the other). From the ammeter, draw a line to the light bulb symbol. From the other terminal of the light bulb, draw a line back to the negative terminal of the battery, completing the main loop. To measure the potential difference across the bulb, draw lines from the two terminals of the bulb and connect them to the two terminals of the voltmeter symbol, placed in parallel across the bulb.

Example of a simple electric circuit diagram

Ohm’S Law ($ V = IR $)

Ohm's Law is one of the most fundamental and widely used laws in electrical circuits. It describes the relationship between the potential difference applied across a conductor and the electric current flowing through it. This empirical law was formulated by German physicist Georg Simon Ohm.

The law states:

"At constant temperature, the electric current flowing through a conductor is directly proportional to the potential difference across its ends."

Mathematically, this can be expressed as:

$ I \propto V $ (at constant temperature and other physical conditions)

Or, equivalently:

$ V \propto I $ (at constant temperature and other physical conditions)

Introducing a constant of proportionality, we get the familiar equation:

$ V = R I $

Here, $V$ is the potential difference across the conductor, $I$ is the current flowing through it, and $R$ is the constant of proportionality known as the resistance of the conductor.

The resistance $R$ is a measure of the opposition offered by the conductor to the flow of electric current. Its value depends on the material of the conductor, its dimensions (length and cross-sectional area), and its temperature.

From the equation $V = IR$, we can also express current and resistance as:

$ I = \frac{V}{R} $

This form shows that for a given potential difference, the current is inversely proportional to the resistance. Higher resistance means lower current.

$ R = \frac{V}{I} $

This form defines resistance as the ratio of the potential difference across the conductor to the current flowing through it.

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Units of Resistance

The SI unit of resistance is the ohm ($\Omega$), named after Georg Simon Ohm.

One ohm is defined as the resistance of a conductor through which a current of 1 ampere flows when a potential difference of 1 volt is applied across its ends.

$ 1 \Omega = \frac{1 \text{ V}}{1 \text{ A}} $

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Ohmic vs. Non-Ohmic Conductors

Conductors that obey Ohm's Law, i.e., those for which the V-I relationship is linear (a straight line passing through the origin when V is plotted against I) and resistance R is constant under constant physical conditions, are called Ohmic conductors. Most metallic conductors like copper, aluminium, etc., exhibit ohmic behaviour over a wide range of temperatures and voltages.

Materials and devices for which the V-I relationship is not linear or for which the resistance changes with voltage, current, or the direction of current are called non-ohmic conductors. We will discuss their limitations in a later section.

Electric Currents In Conductors

Let's delve deeper into the mechanism of current flow in conductors, particularly in metals, from a microscopic perspective. This understanding helps in appreciating the origin of resistance and Ohm's Law.

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Free Electron Model in Metals

In metallic solids, the outer electrons of the atoms are not bound to individual atoms but are relatively free to move throughout the entire volume of the metal. These are called free electrons or conduction electrons. The remaining part of the atom, consisting of the nucleus and the inner electrons, forms positive ions, which are fixed in a regular crystalline structure (the lattice).

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Random Motion (No Electric Field)

In the absence of any applied potential difference or electric field, these free electrons are in constant, random motion due to their thermal energy. They move in all directions, colliding frequently with the fixed positive ions of the lattice. The speeds of these random thermal motions are quite high, typically around $10^6 \, m/s$ at room temperature. However, since the motion is random, the average velocity of the electrons in any particular direction is zero. Consequently, there is no net flow of charge across any section of the conductor, and hence, no electric current.

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Drift Velocity (With Electric Field)

When a potential difference is applied across the conductor, an electric field ($\vec{E}$) is set up inside it. This electric field exerts a force ($\vec{F} = -e\vec{E}$) on each free electron, directing them opposite to the field (since the charge $e$ is negative). This force attempts to accelerate the electrons in one direction.

However, as the electrons accelerate, they continue to collide with the lattice ions. These collisions scatter the electrons and randomise their direction of motion. The effect of the electric field is not to produce a large acceleration, but rather to give the electrons a small average velocity in the direction opposite to the field, superimposed on their large random thermal motion. This small average velocity is called the drift velocity ($v_d$).

The drift velocity is much smaller than the thermal velocity, typically in the order of $10^{-4} \, m/s$ for a copper wire carrying a current of a few amperes.

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Relaxation Time and Average Acceleration

Let's consider the motion of an electron between two successive collisions. Suppose the velocity of an electron just after a collision is $\vec{u}_i$ (random initial velocity). The force on the electron due to the electric field is $\vec{F} = -e\vec{E}$, and its acceleration is $\vec{a} = \vec{F}/m = -e\vec{E}/m$, where $m$ is the mass of the electron.

If the time elapsed since the last collision is $t$, the velocity of the electron just before the next collision will be $\vec{v}_i = \vec{u}_i + \vec{a}t = \vec{u}_i - \frac{e\vec{E}}{m}t$.

The drift velocity is the average velocity of all electrons. Averaging over all electrons:

$ \vec{v}_d = \langle \vec{v}_i \rangle = \langle \vec{u}_i - \frac{e\vec{E}}{m}t \rangle = \langle \vec{u}_i \rangle - \frac{e\vec{E}}{m} \langle t \rangle $

Since the initial thermal velocities $\vec{u}_i$ are random, their average $\langle \vec{u}_i \rangle$ is zero. The average time $\langle t \rangle$ between collisions is called the average collision time or relaxation time ($\tau$).

Therefore, the drift velocity is:

$ \vec{v}_d = -\frac{e\vec{E}}{m}\tau $

This equation shows that the drift velocity is proportional to the applied electric field.

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Current Density and Mobility

Current density ($\vec{J}$) is a vector quantity defined as the current per unit area flowing through a conductor. Its direction is the direction of current flow.

$ \vec{J} = \frac{I}{A} \hat{n} $ (where $\hat{n}$ is the unit vector in the direction of current)

We derived the relationship between current and drift velocity as $I = neAv_d$. Using current density, we get:

$ J = \frac{I}{A} = ne v_d $ (for magnitude)

Replacing $v_d = (eE/m)\tau$ (magnitude of drift velocity):

$ J = ne \left(\frac{eE}{m}\tau \right) = \frac{ne^2\tau}{m} E $

This equation relates current density to the electric field. The term $\frac{ne^2\tau}{m}$ is a constant for a given material at a specific temperature and is called the electrical conductivity ($\sigma$).

$ \sigma = \frac{ne^2\tau}{m} $

So, the equation becomes:

$ J = \sigma E $

In vector form, since $\vec{J}$ and $\vec{E}$ are in the same direction (for conventional current), we have:

$ \vec{J} = \sigma \vec{E} $

This is another form of Ohm's Law, relating current density and electric field.

Mobility ($\mu$) of a charge carrier is defined as the magnitude of the drift velocity per unit electric field strength.

$ \mu = \frac{|v_d|}{|E|} = \frac{(eE/m)\tau}{E} = \frac{e\tau}{m} $

The conductivity can also be written in terms of mobility:

$ \sigma = ne \mu $

The unit of current density is $A/m^2$. The unit of conductivity is $(\Omega \cdot m)^{-1}$ or Siemens per meter (S/m). The unit of mobility is $m^2/(Vs)$.

Ohm’S Law ($ V = IR $)

We have already introduced Ohm's Law empirically as $V=IR$. Now, let's formally derive this macroscopic form from the microscopic picture using the concepts of current density and conductivity discussed above.

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Derivation of $V=IR$ from $\vec{J} = \sigma \vec{E}$

Consider a conductor of length $L$ and uniform cross-sectional area $A$. Let a potential difference $V$ be applied across its ends. Assuming the electric field inside the conductor is uniform, its magnitude is $E = V/L$. The electric field points from the higher potential end to the lower potential end.

The magnitude of the current density in the conductor is $J = I/A$.

From the microscopic form of Ohm's Law, $\vec{J} = \sigma \vec{E}$, the magnitudes are related by $J = \sigma E$.

Substitute $J = I/A$ and $E = V/L$ into this equation:

$ \frac{I}{A} = \sigma \left(\frac{V}{L}\right) $

Rearrange the equation to solve for $V$:

$ V = I \left(\frac{L}{\sigma A}\right) $

Compare this equation with the empirical Ohm's Law $V = IR$. We see that the resistance $R$ of the conductor is given by:

$ R = \frac{L}{\sigma A} $

The reciprocal of conductivity ($\sigma$) is called resistivity ($\rho$), which represents the intrinsic opposition of a material to current flow.

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

Substituting $\sigma = 1/\rho$ into the equation for $R$:

$ R = \rho \frac{L}{A} $

This is the expression for the resistance of a conductor in terms of its material property (resistivity $\rho$) and its geometry (length $L$ and cross-sectional area $A$).

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Resistivity and Factors Affecting Resistance

Resistivity ($\rho$) is a characteristic property of the material at a given temperature. It does not depend on the dimensions of the conductor. Low resistivity indicates that a material is a good conductor (e.g., metals), while high resistivity indicates a poor conductor or insulator. The unit of resistivity is ohm-metre ($\Omega \cdot m$).

From $R = \rho L/A$, the resistance of a conductor is:

• Directly proportional to its length ($L$). A longer wire offers more resistance.
• Inversely proportional to its cross-sectional area ($A$). A thicker wire offers less resistance.
• Depends on the material of the conductor ($\rho$). Different materials have different resistivities.
• Depends on the temperature. For most metallic conductors, resistance and resistivity increase with increasing temperature because the thermal vibrations of the lattice ions become more vigorous, leading to more frequent collisions and reduced relaxation time $\tau$.

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V-I Characteristics for Ohmic Conductors

For an ohmic conductor, the resistance $R$ is constant at a constant temperature. The relationship $V=IR$ implies that the potential difference ($V$) is linearly proportional to the current ($I$). If we plot a graph with $I$ on the x-axis and $V$ on the y-axis, the graph is a straight line passing through the origin. The slope of this $V$-$I$ graph is equal to the resistance $R$. If we plot $V$ on the x-axis and $I$ on the y-axis, the graph is also a straight line through the origin, but its slope is $1/R$, which is the conductance ($G$).

V-I graph for an Ohmic conductor. Slope = $V/I = R$.

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Answer:

Limitations Of Ohm’S Law

Ohm's Law, $V=IR$, is a powerful tool for analysing many electrical circuits, particularly those involving metallic conductors at constant temperature. However, it is important to recognise that Ohm's Law is an empirical relation, not a fundamental law derived from first principles that applies to all materials under all conditions. There are several cases where the proportionality between voltage and current does not hold, and the resistance $R=V/I$ is not a constant. These materials or devices are called non-ohmic.

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Conditions for Ohm's Law Applicability

Ohm's Law strictly applies when:

1. The material is Ohmic.
2. The temperature of the conductor is kept constant. (The temperature of a conductor can change significantly due to the heating effect of current, especially at higher current values).
3. Other physical conditions, such as pressure, illumination (for certain materials like photodiodes), etc., are kept constant.

When these conditions are not met, Ohm's Law may not be valid.

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Common Cases Where Ohm's Law Fails

Here are the primary ways in which the V-I relationship for a material can deviate from Ohm's Law, leading to non-ohmic behaviour:

1. Non-linear V-I Characteristic: The most common deviation is when the current is not directly proportional to the applied voltage. The V-I graph is a curve, not a straight line. In such cases, the ratio $R=V/I$ is not constant; it changes with voltage or current.

   Example: Filament lamp. As current flows through the filament, it heats up considerably. The resistance of metals increases with temperature ($R \propto T$). So, as the voltage across the lamp increases, the current increases, which increases the temperature, which in turn increases the resistance. The increase in current is less than proportional to the increase in voltage at higher voltages. The V-I graph will be curved upwards.

   V-I graph for a filament lamp. The slope ($R$) increases with V.

2. V-I Relationship Depends on the Sign of V: For certain devices, the relationship between V and I depends on the polarity of the applied voltage. The magnitude of the current for a given voltage might be different when the voltage is reversed.

   Example: Semiconductor diode (p-n junction diode). A diode conducts current easily when forward-biased (positive voltage applied to the p-side, negative to the n-side), but conducts very little current when reverse-biased (polarity reversed), even for significant voltages. The V-I graph is very different in the forward and reverse bias regions.

   V-I graph for a p-n junction diode. Current flows significantly only in forward bias.

3. Non-unique V-I Relationship: In some materials, the same value of current might correspond to more than one value of voltage, or vice versa. This is often observed in materials exhibiting negative resistance regions.

   Example: Gallium Arsenide (GaAs). For certain ranges of voltage, increasing the voltage can actually lead to a decrease in current. This behaviour, known as negative differential resistance, results in a V-I characteristic where the curve may loop back on itself for certain voltage ranges.

   V-I graph for Gallium Arsenide, showing a non-unique current value for a given voltage (in the dip).

In summary, while Ohm's Law provides a simple and effective model for many basic circuit analyses, especially for metals under stable conditions, it is crucial to be aware of its limitations and the behaviour of non-ohmic materials encountered in modern electronic devices and systems.