Across a metallic conductor of non-uniform cross section a constant potential difference is applied. The quantity which remains constant along the conductor is:(A) Drift velocity(B) Electric field(C) Current density(D) Current

To determine which quantity remains constant along a metallic conductor of non-uniform cross-section when a constant potential difference is applied, let's analyze each option.
Step 1: Understanding the Given Scenario
• A metallic conductor has a non-uniform cross-sectional area, meaning the area of the conductor varies along its length.
• A constant potential difference (V) is applied across the conductor.
• The conductor follows Ohm's Law and the equation of continuity.
Step 2: Analyzing Each Option
(A) Drift Velocity (vdv_d)
Drift velocity is given by:
vd=IneAv_d = \frac{I}{n e A}
where:
• II is the current,
• nn is the number of charge carriers per unit volume,
• ee is the charge of an electron,
• AA is the cross-sectional area.
Since the cross-sectional area (AA) is varying along the conductor, the drift velocity (vdv_d) must also vary to maintain a constant current.
✅ Drift velocity is NOT constant.
(B) Electric Field (EE)
Ohm's Law states:
E=ρJE = \rho J
where ρ\rho is the resistivity and JJ is the current density. Since the conductor has a non-uniform cross-section, the resistance varies along the conductor, causing the electric field to change.
✅ Electric field is NOT constant.
(C) Current Density (JJ)
Current density is given by:
J=IAJ = \frac{I}{A}
Since the cross-sectional area AA varies along the conductor, but the current II remains constant, the current density JJ must change accordingly.
✅ Current density is NOT constant.
(D) Current (II)
According to the principle of continuity, for a steady-state condition:
I=JAI = J A
Since charge cannot accumulate at any point in a conductor, the current remains constant throughout despite variations in cross-sectional area.
✅ Current is CONSTANT.
The quantity that remains constant along the conductor is:
(D) Current