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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.
Solution
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.
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