Electrical Resistance

Electrical resistance is the characteristic of a material that resists the flow of electric current. It can be compared to how a narrow pipe resists a flow of water inside it. When a voltage or electric potential difference is applied across a material, the electrons in this material move. 

1.0Definition of Electrical Resistance 

Electrical resistance is the property of a material that opposes the flow of electric current. To get a clear idea, imagine a water pipe, a narrow pipe takes more time to fill a tank in comparison to a wider pipe. Similarly, when voltage, or electric potential difference, is applied across a material, the electrons inside move. This resistance determines how much the material resists the movement of those electrons. The unit of resistance is the Ohm (Ω), named after German physicist Georg Simon Ohm, who formulated Ohm's Law.

2.0Ohm’s Law

One of the most important concepts in Electrical resistance physics is Ohm’s law. It is the mathematical representation of the relation between current, resistance, and voltage. The current (I) flowing from a conductor is always directly proportional to the voltage (V) applied across it and inversely to the resistance of the conductor. The relation is given by:

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

Where:

• V = voltage (in volts, V),
• I = current (in amperes, A),
• R = resistance (in ohms, Ω).

This means that for a given voltage if the resistance increases, the current decreases, & vice versa. Ohm’s Law is fundamental to understanding how electrical circuits behave.

3.0Electrical Resistance and Resistivity

Resistivity is the quantification of a property of a material, determining how strongly it resists the flow of electric current. This is similar to resistance but intrinsic to the material- it does not depend on the shape or size of the material. Resistivity (ρ) measured in ohm-meters (Ω·m) and given by the formula:

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

Where:

• R is the resistance (in ohms, Ω),
• 𝜌 is the resistivity of the material (in ohm-meters, Ω·m),
• L is the length of the material (in meters, m),
• A is the cross-sectional area (in square meters, m²).
• The dimensional formula for resistivity 𝜌 is [ML³T^-3I^-2] and resistance is [R] = [ML²T^-3I^-2]

This formula helps to understand how the resistance of a conductor is related to the material's resistivity, length, and cross-sectional area.

4.0Electric Resistance and Temperature

Electric Resistance and temperature are related to each other by the temperature coefficient of resistance of a material (primarily conductors, semiconductors, and insulators). The temperature coefficient of resistance, commonly referred to as α, measures the rate of change in the resistance of a material due to a temperature change. The coefficient is defined as the fractional change in resistance for a given change in temperature. Mathematically temperature coefficient is expressed as:  

$R_T=R_0(1+\alpha (T-T_0))$

Here:

• R_T = Resistance at temperature T
• R₀ = Resistance at a reference temperature T₀​ (often taken as 20°C)
• α = Temperature coefficient of resistance (in per degree Celsius, °C⁻¹)
• T = Temperature at which resistance is measured (in °C)
• T₀​ = Reference temperature (in °C)

The electric resistance of different materials varies with different temperatures: 

Conductors:

• The resistance is directly proportional to the temperature, meaning as the temperature increases in a conductor and then, the resistance of the conductor also increases. This happens because as the temperature increases, the atoms of the conductor vibrate more violently making it difficult for free electrons to pass through easily. 
• The temperature coefficient of resistance of conductors is positive, meaning the resistance increases with an increase in temperature.

Insulators:

• The electrons in insulators are tightly bound to their atoms and do not flow freely; therefore, they have high resistance. When the temperature of an insulator is increased, it makes the thermal energy cause more vibrations in the atoms, which further hinders any small amount of free charge that might be present for moving; hence, the resistance of insulators increases with an increase in temperature. 
• Insulators (such as rubber, wood, and ceramics) also have a positive temperature coefficient. However, they are usually very high in resistance to start with, and the effect of temperature on resistance is less noticeable until extremely high temperatures are reached.

Semiconductors:

• In semiconductors, when the temperature rises, the resistance reduces. At high temperatures, more electrons gain energy to break free from their atoms, increasing the free charge carriers that help conduct electricity. For this reason, most devices for transistors and diodes use semiconductors since controlling the resistance through temperature variations is a significant task in these devices.
• Semiconductors (like silicon, germanium, and gallium arsenide) have a negative temperature coefficient. This means their resistance decreases with an increase in temperature.

5.0Electrical Resistance Examples

• Electrical Resistance of Copper Wire: The extensive use of copper in electrical wires is because of its exceptional conductivity. The resistance of the copper wire is very low, which helps reduce energy consumption and ensures the efficient functioning of circuits. 
• Electrical Resistance of Tungsten: Although highly resistive by comparison to copper, tungsten is still used in conditions requiring high temperatures, such as in light bulbs. Tungsten has a resistance that peaks with increasing temperatures. Thus, this material is used in applications requiring heating elements.
• Electrical Resistance of the Human Body: The human body's resistance is different under different conditions, such as skin moisture and contact points. Dry skin has greater resistance than wet skin. Exposition of a man to electrical currents will depend on how much current flows in it, based on the resistance of the body. The higher the resistance, the less the current flowing, lowering the risk of electrical shock.
• The Electric Resistance of a Stretched Wire: According to the formula of resistance, resistance is directly proportional to the length of the wire and inversely to the cross-sectional area of the wire. By stretching the wire the length of the wire is increased, as well as the cross-sectional area is decreased. Hence, the resistance of the wire increases if stretched.

6.0Solved Examples

Problem 1: In a Wheatstone bridge, the resistances are as follows: R₁=100 Ω, R₂=200 Ω, and R₃ = 300 Ω. If the bridge is balanced, calculate the value of R₄.

Solution: The balanced condition for a Wheatstone bridge is given by: 

$\frac{R_1}{R_2}=\frac{R_3}{R_4}$

Substituting the values of Resistances: 

$\frac{100}{200}=\frac{300}{R_4}$

$R_4=\frac{300\times 200}{100}=600\Omega$

Problem 2: The resistance of a conductor at 0^∘C is R₀=10 Ω. The temperature coefficient of resistance for the material is α=0.004 °C⁻¹. Find the resistance of the conductor at 50^∘C.

Solution: The resistance of a conductor changes with temperature according to the following equation: 

$R_T=R_0(1+\alpha (T-T_0))$

Given that R₀= 10Ω, α=0.004 °C⁻¹, T₀ = 0^∘C, T = 50^∘C

R_T = 10(1+0.004(50–0))

R_T = 10(1+0.004(50))

R_T = 10(1+0.2) = 10(1.2) 

R_T = 12

Problem 3: A wire has a length of L=2.0 m and is made of a material with resistivity ρ=1.7×10⁻⁸ Ω.m. The radius of the wire is r = 0.5 mm. Find the resistance of the wire. (use $\pi$= 3.1416)

Solution: For the resistance of a wire, we have the formula: 

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

Given that L=2.0 m, r = 0.5 mm = 0.510^-3m, ρ=1.7×10⁻⁸ Ω.m

So, to find the cross-sectional area, we have = $\pi r^2$

$=\pi (0.5\times 10^{-3}{)}^2$

= 3.1416 $\times$0.25 $\times$$10^{-6}$

$=0.7854\times 10^{-6}{m}^2$

$A=7.854\times 10^{-7}{m}^2$

Now put these values in the above formula: 

$R=(1.7\times 10^{-8})\times \frac{2.0}{7.854\times 10^{-7}}$

$R=(1.7\times 10^{-8})\times (2.544\times 10^6)$

$R=4.33\times 10^{-2}\Omega =0.0433\Omega$