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:

$V=I\times R$

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).

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

The electric resistance of different materials varies with different temperatures: 

1. 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. 
2. Insulators: Insulators tend to exhibit reduced resistivity at higher temperatures and, consequently, some degree of conductivity. This is because any heat applied to the insulator can excite its electrons, making it easier for them to move across this material and conduct electricity.
3. 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.

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.

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: 

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

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

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