Question

Two conductors, made of the same material have equal lengths but different cross-sectional areas A₁ and A₂ (A₁ > A₂). They are connected in parallel across a cell. Show that the drift velocities of electrons in two conductors are equal.

Answer 1

When two wires are connected in parallel the potential difference across the wires will be the same.

V₁ = V₂ ...............(i)

V₁ = Potential across wire A.

V₂ = Potential across wire B.

Since two conductors are made up of the same material and have equal lengths, their resistivity will be the same,

`rho_1 = rho_2 = rho`

and `(R_2)/(R_1) = A_1/A_2` ........(ii)

Current (I₁) flowing through the conductor with cross-sectional area A₁ is given by

`I_1 = A_1n ev_1`

and `I_2 = A_2n ev_2` ..............(iii)

Since two conductors are connected in parallel.

I = I₁ + I₂

Using Ohm's law,

V = IR

From (i) I₁R₁ = I₂R₂

From (iii)

`A_1n ev_1R_1 = A_2n ev_2R_2`

`(A_1v_1)/(A_2v_2) = R_2/R_1`

From (ii)

`A_1/A_2 v_1/v_2 = A_1/A_2`

`v_1 = v_2`

Hence, the drift velocities of electrons in the two conductors are equal.