Question

Two wires made of the same material have the same length \( l \) but different cross-sectional areas \( A_1 \) and \( A_2 \). They are connected together with a cell of voltage \( V \). Find the ratio of the drift velocities of free electrons in the two wires when they are joined in: (i) series, and (ii) parallel.

Hint:

In series circuits, current is constant, so drift velocity varies inversely with cross-sectional area. In parallel circuits, voltage is constant, and current divides in proportion to area, keeping drift velocities equal.

Answer 1

The drift velocity of electrons is defined as:\[v_d = \frac{I}{nAe}\]with the following parameters: \( v_d \) representing drift velocity, \( I \) representing current, \( n \) representing the number density of electrons, \( A \) representing cross-sectional area, and \( e \) representing the elementary charge.(i) Series Connection:In a series circuit, the current \( I \) remains constant across all components.Consequently, the drift velocities are expressed as:\[v_{d1} = \frac{I}{nA_1e}, \quad v_{d2} = \frac{I}{nA_2e}This leads to the ratio:\[\Rightarrow \frac{v_{d1}}{v_{d2}} = \frac{A_2}{A_1}\](ii) Parallel Connection:In a parallel circuit, the voltage \( V \) is identical across each branch.Given the resistance formula \( R = \frac{\rho l}{A} \), current is directly proportional to the cross-sectional area (\( I \propto A \)).Therefore:\[I_1 \propto A_1, \quad I_2 \propto A_2\]The drift velocities are calculated as:\[v_{d1} = \frac{I_1}{nA_1e} = \frac{A_1}{nA_1e} = \frac{1}{ne}\]\[v_{d2} = \frac{I_2}{nA_2e} = \frac{A_2}{nA_2e} = \frac{1}{ne}Resulting in the ratio:\[\Rightarrow \frac{v_{d1}}{v_{d2}} = 1\]