# Why is current same even if the area of the conductor differs?

Current is directly proportional to area if area of conductor increase Current increases but in tapering conductor area increases current remains same

I have learned this equation I=enAv Where A is area and v is drift velocity,as area increases does drift velocity decrease so that the current remains constant,does the density at smaller area increases which leads to increase in drift velocity or current remains constant with change in area as number of electron passing through cross section remains same at any velocity?

• Due to surface charges, if 2 wires are connected then the electric field value is different, making it constant Commented May 19, 2022 at 17:25

## 1 Answer

(a) "Current is directly proportional to area" In general it isn't. It will be the same, though, when we connect the same voltage across wires of different cross-sectional area, but of the same length and material. In this case, $$n$$ and $$v$$ (and, of course, $$e$$) will be the same in the equation $$I=nAve$$.

(b) Within a very short time of applying a constant voltage across conductors of different cross-sectional area in series, or across a tapering conductor (your 'frustum'?) the current (rate of flow of charge) will be the same throughout the composite or tapering conductor. If it were't constant, charge (positive or negative) would continue to build up in places along the conductor which couldn't happen because of mutual repulsion between charges preventing further charge building up.

(c) In the composite or tapering conductor (assumed homogeneous in material) the drift velocity $$v$$ adjusts to be greater where the conductor is thinner, so $$vA$$ is the same all along the conductor.

• If the velocity of electron increasing according to area can we say it is accelerating which leads to lose of energy? Commented May 20, 2022 at 4:40
• Remember that $v$ is the so-called drift velocity, the mean rate of travel of free electrons in the direction of the applied electric field. It is superimposed on the much greater random thermal velocity of the free electrons. This means that the increase in drift velocity due to decrease in area of conductor makes only a very small change in the electron's velocity. I think that you $could$ regard it as an acceleration, but I'm on shaky ground because the motion of free electrons in a metal is strongly influenced by the forces between the electrons and the metal lattice. Commented May 20, 2022 at 15:45
• Regarding energy, I think that any increase in free electrons' kinetic energy due to decrease in the conductor's cross-section would be negligible compared to the energy transferred to random thermal energy in the collisions that are place continually between the free electrons and the lattice of ions. Commented May 20, 2022 at 15:48