Electrical Resistance

Imagine water flowing through a pipe. If the pipe is narrow or has obstacles, the water flow is restricted. Similarly, in an electrical circuit, the flow of electric charge (current) can be restricted by the material it flows through. This opposition to the flow of current is called electrical resistance.

Resistance is a fundamental property of a material that determines how easily electric current can pass through it. It's like friction for electricity. Materials with high resistance are poor conductors of electricity, while materials with low resistance are good conductors.

Factors Affecting Resistance

The resistance of a conductor depends on several factors:

• Length of the conductor (L): The longer the conductor, the more opposition there is to the flow of charge. Think of a long pipe versus a short pipe for water – the longer pipe offers more resistance. So, resistance is directly proportional to the length of the conductor.

  R ∝ L

• Area of cross-section (A): A thicker conductor (larger cross-sectional area) allows more charge to flow through it easily. Imagine a wider pipe for water – it allows more water to flow. So, resistance is inversely proportional to the area of cross-section.

  R ∝ 1/A

• Nature of the material: Different materials have different inherent abilities to resist the flow of charge. Metals like copper and silver are excellent conductors with very low resistance, while materials like rubber or plastic are insulators with very high resistance. This property is quantified by a factor called resistivity.
• Temperature: For most conductors, resistance increases as temperature increases. This is because at higher temperatures, atoms vibrate more vigorously, colliding with the flowing electrons and hindering their movement. For semiconductors and insulators, the effect of temperature can be different.

Resistivity (ρ)

By combining the effects of length and area, we can write the relationship for resistance as:

R = ρ (L/A)

Here, 'ρ' (rho) is the resistivity of the material. Resistivity is an intrinsic property of a material, independent of its shape or size. It tells us how strongly a material opposes the electric current.

The unit of resistivity is Ohm-meter (Ω·m). A material with low resistivity is a good conductor, and a material with high resistivity is a poor conductor or an insulator.

Ohm's Law and Resistance

Ohm's Law is a fundamental principle that relates voltage, current, and resistance in a circuit. It states that the current flowing through a conductor is directly proportional to the voltage applied across its ends, provided the temperature and other physical conditions remain unchanged.

V ∝ I

V = IR

Where:

• V is the voltage (potential difference) across the conductor (in Volts, V).
• I is the current flowing through the conductor (in Amperes, A).
• R is the resistance of the conductor (in Ohms, Ω).

From Ohm's Law, we can also express resistance as:

R = V/I

This means resistance is the ratio of the voltage applied to the conductor to the current flowing through it. A higher resistance value means that a larger voltage is required to drive the same amount of current, or for a given voltage, a smaller current will flow.

Units of Resistance

The SI unit of resistance is the Ohm (Ω), named after Georg Simon Ohm.

• 1 Ohm (Ω) is defined as the resistance of a conductor when a potential difference of 1 Volt (V) applied across it produces a current of 1 Ampere (A).
• 1 Ω = 1 V / 1 A

Larger units include kilohms (kΩ = 10³ Ω) and megohms (MΩ = 10⁶ Ω).

Example: If a resistor has a resistance of 100 Ω, it means that for every 100 Volts applied across it, 1 Ampere of current will flow through it.

V-I Characteristics of Ohmic and Non-Ohmic Conductors

The relationship between the voltage (V) applied across a conductor and the current (I) flowing through it can be represented graphically. This graph is called the V-I characteristic or V-I curve. The shape of this curve tells us whether the conductor obeys Ohm's Law or not.

Ohmic Conductors

Ohmic conductors are those materials or devices that obey Ohm's Law. For these conductors, the resistance remains constant irrespective of the applied voltage or the current flowing through them, as long as the temperature and other physical conditions are kept constant.

Characteristics:

• The V-I graph for an ohmic conductor is a straight line passing through the origin.
• The slope of the V-I graph (V/I) represents the resistance (R), which is constant.
• Examples include metallic conductors like copper, silver, aluminum wires, and resistors specifically designed to have constant resistance.

Let's consider an example. If we apply 2V across a resistor and get 1A current, its resistance is 2Ω. If we increase the voltage to 4V, the current becomes 2A, and the resistance is still 4V/2A = 2Ω. If we further increase the voltage to 6V, the current becomes 3A, and the resistance remains 6V/3A = 2Ω. The resistance is constant.

V-I Graph for Ohmic Conductor:

(Imagine a graph with Voltage on the Y-axis and Current on the X-axis. It's a straight line originating from (0,0) and sloping upwards.)

Non-Ohmic Conductors

Non-ohmic conductors are those materials or devices that do not obey Ohm's Law. For these conductors, the resistance is not constant; it changes with the applied voltage or the current. The V-I graph for a non-ohmic conductor is not a straight line.

Characteristics:

• The V-I graph is a curve, not a straight line.
• The resistance (V/I) varies depending on the point on the curve.
• The resistance at any point on the curve can be found by calculating the ratio V/I for that specific point, or by considering the dynamic resistance (dV/dI) which is the slope of the tangent to the curve at that point.

Examples of Non-Ohmic Devices:

• Diodes: These are semiconductor devices that allow current to flow primarily in one direction. Their V-I characteristic is highly non-linear.
• Transistors: Used for amplification and switching, their behavior is non-ohmic.
• Filament lamps: As the filament heats up due to the current, its resistance increases. So, the V-I graph is a curve bending upwards.
• Thermistors: Their resistance changes significantly with temperature, leading to non-ohmic behavior in many applications.

Let's consider a filament lamp. At low voltages, it behaves somewhat like an ohmic conductor. However, as voltage increases, the filament gets hotter, and its resistance increases. This means that for a further increase in voltage, the increase in current is less than what would be expected from Ohm's Law.

V-I Graph for Non-Ohmic Conductor (e.g., Filament Lamp):

(Imagine a graph with Voltage on the Y-axis and Current on the X-axis. It starts as a straight line from the origin but then curves upwards, indicating increasing resistance.)

Electrical Energy and Power

When electric current flows through a conductor, electrical energy is transferred. This energy can be converted into other forms, such as heat, light, or mechanical work. Electrical power is the rate at which this energy is transferred or converted.

Electrical Energy (E)

Electrical energy is the work done by the electric field to move charges through a conductor. It is the total amount of energy transferred by an electric circuit per unit time.

We know that voltage (V) is the work done per unit charge (V = W/q).

Current (I) is the charge flowing per unit time (I = q/t).

Therefore, charge q = I × t.

Substituting q in the voltage formula: V = W / (I × t)

So, Work done (W), which is equal to the electrical energy transferred (E), is:

E = W = V × I × t

The unit of electrical energy is the Joule (J).

Using Ohm's Law (V = IR), we can express electrical energy in other forms:

• Substituting V = IR: E = (IR) × I × t = I²Rt
• Substituting I = V/R: E = V × (V/R) × t = V²t / R

So, the formulas for electrical energy are:

E = VIt = I²Rt = V²t / R

Example: If a 12V battery is connected to a 6Ω resistor for 10 seconds, how much energy is transferred? First, find the current: I = V/R = 12V / 6Ω = 2A. Now, calculate energy: E = VIt = 12V × 2A × 10s = 240 Joules. Alternatively, E = I²Rt = (2A)² × 6Ω × 10s = 4 × 6 × 10 = 240 Joules. Or, E = V²t / R = (12V)² × 10s / 6Ω = 144 × 10 / 6 = 24 × 10 = 240 Joules.

Electrical Power (P)

Electrical power is the rate at which electrical energy is transferred or converted. It is the amount of work done per unit time.

Power (P) = Energy (E) / Time (t)

Substituting the formula for energy (E = VIt):

P = (VIt) / t = VI

The SI unit of power is the Watt (W). 1 Watt is equal to 1 Joule per second (1 W = 1 J/s).

Using Ohm's Law (V = IR), we can derive other formulas for power:

• Substituting V = IR: P = (IR) × I = I²R
• Substituting I = V/R: P = V × (V/R) = V² / R

So, the formulas for electrical power are:

P = VI = I²R = V² / R

Example: A heater element has a resistance of 50Ω and operates on a 200V supply. Calculate the power consumed by the heater. Using P = V² / R: P = (200V)² / 50Ω = 40000 V² / 50Ω = 800 Watts.

If the current drawn by the heater is required, we can find it using P = VI: I = P / V = 800W / 200V = 4 Amperes. Or using Ohm's law directly: I = V/R = 200V / 50Ω = 4 Amperes.

Commercial Unit of Energy: The unit of electrical energy used in homes and industries is the kilowatt-hour (kWh), often called a "unit" of electricity.

1 kWh = 1 kilowatt × 1 hour 1 kWh = 1000 Watts × 3600 seconds 1 kWh = 3,600,000 Joules = 3.6 × 10⁶ J

Example: If you use a 100W bulb for 10 hours, the energy consumed is: Energy = Power × Time = 100W × 10 hours = 1000 Wh = 1 kWh. This means 1 kWh of energy has been transferred.