Correct Answer - Option 1 : One fourth

__CONCEPT:__

Resistance:

- The property of any conductor that opposes the flow of electric current through it and depends on the shape and size of the materials, temperature, and nature of the materials is called resistance.
- It is denoted by R and the SI unit is the ohm (Ω).

- The resistance is given by:

\(⇒ R=\frac{ρ l}{A}\)

Where ρ = specific resistance, l = length of the wire, and A = cross-sectional area of the wire

Ohm's law

- Ohm’s law states that the voltage across a conductor is directly proportional to the current flowing through it, provided all physical conditions and temperature remain constant.

⇒ V = IR

Where V = voltage, I = current and R = resistance

__CALCULATION:__

Given l_{1} = l_{2} = l, A_{1} = A_{2} = A and ρ_{2} = 4ρ_{1}

For case 1, (ρ_{1} = ρ)

\(⇒ R_{1}=\frac{ρ l}{A}\) -----(1)

The current in the circuit,

\(\Rightarrow I_{1} = \frac{V}{R_{1}}\)

\(\Rightarrow I_{1} = \frac{VA}{\rho l}\) -----(2)

For case 2, (ρ2 = 4ρ)

\(⇒ R_{2}=\frac{4ρ l}{A}\) -----(3)

The current in the circuit,

\(\Rightarrow I_{2} = \frac{V}{R_{2}}\)

\(\Rightarrow I_{2} = \frac{VA}{4\rho l}\) -----(4)

By equation 2 and equation 4,

\(\Rightarrow I_{2} = \frac{I_{1}}{4}\)

- Hence, option 1 is correct.