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Step-by-Step Explanation
1. Introduction
When a constant potential difference is applied across a metallic conductor, charges (electrons) flow to constitute an electric current. Even if the conductor’s cross-sectional area varies, certain quantities of interest may change along the conductor, while others remain constant.
2. Key Quantities Involved
Drift Velocity ($v_d$): The average velocity acquired by free electrons due to the electric field within the conductor. If the cross section changes, the local electric field and hence drift velocity can vary along the conductor.
Current Density ($J$): Defined as $J = \frac{I}{A}$, where $I$ is the current and $A$ is the cross-sectional area. When the cross section is non-uniform, $J$ will change because $A$ changes.
Electric Field ($E$): The electric field within the conductor responsible for causing the electrons to move. It may vary with position depending on the potential gradient and conductor geometry.
Current ($I$): The rate at which charge flows through the cross section of the conductor. In a steady state (constant flow of charge), the same current flows through every cross section of a series conductor, regardless of variations in area.
3. Why Current Remains Constant
For a continuous, steady flow of charges, the same amount of charge per unit time must pass through every cross section of the conductor. If different amounts of current flowed in different sections, charge would accumulate somewhere in the conductor, which does not happen in steady-state conditions. Therefore:
Even if the cross-sectional area changes, the net current $I$ passing through all sections must remain the same.
However, the current density $J$ can vary because it depends on the cross-sectional area, and drift velocity may also vary due to the corresponding change in the local electric field.
4. Conclusion
Given a metallic conductor with a non-uniform cross section under a constant potential difference, the only quantity that remains the same throughout the conductor is the current. This is because it is a single continuous path for charge flow and, in steady state, charge cannot build up or vanish inside the conductor, ensuring the same current in every region of the conductor.
