1. What are Dimensional Formulas?

Dimensional formulas express physical quantities in terms of fundamental dimensions: Mass (M), Length (L), Time (T), and Electric Current (I). They help verify equations and convert units between different systems of measurement.

2. Resistivity Dimensional Formula

The dimensional formula for resistivity (ρ) is:

Derivation: Resistivity is defined as ρ = RA/L, where:

• R (Resistance) = [M L² T^{-3} I^{-2}]
• A (Area) = [L²]
• L (Length) = [L]

Therefore: ρ = [M L² T^{-3} I^{-2}] × [L²] / [L] = [M L³ T^{-3} I^{-2}]

3. Conductivity Dimensional Formula

The dimensional formula for conductivity (σ) is:

Derivation: Since conductivity is the reciprocal of resistivity:

• σ = 1/ρ
• Therefore: σ = 1/[M L³ T^{-3} I^{-2}] = [M^{-1} L^{-3} T³ I²]

4. Relationship Between Resistivity and Conductivity

Key Points: Resistivity and conductivity are inversely related. Their dimensional formulas are reciprocals of each other, which confirms their mathematical relationship σ = 1/ρ.

5. Frequently Asked Questions (FAQ)

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

Q2: What do the exponents in dimensional formulas represent?
A: Exponents indicate how the quantity depends on fundamental dimensions. For example, L³ means the quantity depends on volume.

Q3: Can dimensional formulas be used for numerical calculations?
A: No, dimensional formulas only show the dependence on fundamental dimensions, not numerical values.

Q4: What are the SI units for resistivity and conductivity?
A: Resistivity: ohm-meter (Ω·m), Conductivity: siemens per meter (S/m).

Q5: How do temperature changes affect resistivity and conductivity?
A: For most metals, resistivity increases with temperature, so conductivity decreases. The dimensional formulas remain unchanged.