Electrical Resistivity And Conductivity Dimensional Formula

Dimensional Formulas:

\[ [\rho] = [M L^3 T^{-3} I^{-2}] \] \[ [\sigma] = [M^{-1} L^{-3} T^3 I^2] \]

Dimensional Formula:

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1. What Are Electrical Resistivity And Conductivity?

Electrical resistivity (ρ) is a fundamental property that quantifies how strongly a material opposes the flow of electric current. Electrical conductivity (σ) is the reciprocal of resistivity and measures a material's ability to conduct electric current.

2. Understanding Dimensional Formulas

Dimensional formulas express physical quantities in terms of fundamental dimensions:

\[ M = \text{Mass}, \quad L = \text{Length}, \quad T = \text{Time}, \quad I = \text{Electric Current} \]

These fundamental dimensions help verify equations and understand relationships between physical quantities.

3. Dimensional Analysis of Resistivity

The dimensional formula for electrical resistivity is derived from its definition:

\[ \rho = R \frac{A}{L} \] \[ [\rho] = [R] \cdot \frac{[A]}{[L]} = [M L^2 T^{-3} I^{-2}] \cdot \frac{[L^2]}{[L]} = [M L^3 T^{-3} I^{-2}] \]

Where:

\( R \) — Electrical resistance ([M L² T⁻³ I⁻²])
\( A \) — Cross-sectional area ([L²])
\( L \) — Length ([L])

4. Dimensional Analysis of Conductivity

Since conductivity is the reciprocal of resistivity:

\[ \sigma = \frac{1}{\rho} \] \[ [\sigma] = [\rho]^{-1} = [M^{-1} L^{-3} T^{3} I^{2}] \]

This shows the inverse relationship between resistivity and conductivity dimensions.

5. Frequently Asked Questions (FAQ)

Q1: Why are dimensional formulas important?
A: Dimensional formulas help verify the correctness of physical equations, convert units, and understand the nature of physical quantities.

Q2: What are the SI units of resistivity and conductivity?
A: Resistivity: ohm-meter (Ω·m), Conductivity: siemens per meter (S/m).

Q3: How do resistivity and conductivity relate to each other?
A: They are reciprocals: σ = 1/ρ. Materials with high resistivity have low conductivity and vice versa.

Q4: What factors affect electrical resistivity?
A: Temperature, material composition, impurities, and crystal structure all affect a material's resistivity.

Q5: Are these dimensional formulas universal?
A: Yes, these dimensional formulas are valid in all systems of units since they are expressed in fundamental dimensions.