Question

Two metallic rods, each of length L, area of cross A₁ and A₂, having resistivities ρ₁ and ρ₂ are connected in parallel across a d.c. battery. Obtain the expression for the effective resistivity of this combination.

Answer 1

For resistor R₁:

Resistivity = ρ₁

Length = L

Area = A₁

∴ R₁ = `rho_1 xx L/A_1`

For resistor R₂:

Resistivity = ρ₂

Length = L

Area = A₂

∴ R₂ = `rho_2 xx L/A_2`

For the equivalent resistor:

Resistivity = ρ

Length = L

Area (A) = A₁ + A₂

∴ R = `rho xx L/A`

Since the resistors are connected in parallel.

Equivalent resistance (R) = `(R_1 R_2)/(R_1 + R_2)`

Or,

`rho xx L/A = ((rho_1L)/A_1 xx (rho_2L)/A_2)/((rho_1L)/A_1 + (rho_2L)/A_2`

∴ Effective resistivity (ρ) = `(rho_1 rho_2 (A_1 + A_2))/(rho_1 A_2 + rho_2 A_1)`