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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.
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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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