A wire whose cross-sectional area is increasing linearly from its one end to the other, is connected across a battery of V volts. Which of the following quantities remains constant in the wire ?
(a) drift speed , (b) current density
(c ) electric current , (d) electric field.
Justify your answer.

Explanation

(i) Consider an electron of mass m and charge e moving inside a conductor with a drift velocity `V_(d)` when an electric field E is applied.
Force on electron = eE. ma is the mechanical force the electron.
`{:( :.,ma=eE,rArr,a=(eE)/m),("Now",v=u+at,,),("Here",u=0,,a=(eE)/(m)),(,t=tau(" relaxation time"),:.,u_(d)=(eE)/(m)tau):}`
Ohm's law :
`{:( :.,I="ne"Av_(d),:.,I="ne"(AeE)/(m)tau),( :.,I=("ne"^(2)AV)/(ml)tau,:.,R=V/IrArr (ml)/(A" ne"^(2)tau)),( :., V=(ml)/(" ne"^(2)Atau)I,:.,V prop I):}`
Here Resistance `R =(ml)/(" ne"^(2)Atau)`. ltbgt (ii) Current density remains constant in the wire whose cross sectional area is increasing linearly from its one end to the other as ,
Current density `J = (I)/(A)`,
It is current per unit area, i.e, doesn't depend on area of cross sectional .
Drift speed `v_(d) = 1/(A"ne")` Electric field `= J//A`, is electrical conductivity.