A light bulb lights up almost immediately when connected to a battery. The average motion of many electrons can be determined from the equation of motion for a single electron.
Adjust conditions:
The sliders adjust the values
断面積1 mm²の銅線に1 A。電子の平均位置が1 m進む時間は？
実効値1 A・50 Hzの正弦波交流。3.8時間に平均位置が動く範囲は？
The equations explain the phenomenon
1The force exerted by the electric field gives the electron's equation of motion.m dvdt=−e E……
2The average velocity is calculated by accounting for changes in direction caused by collisions.vavg=−e E τm……
3The electric current is calculated from the net number of electrons crossing a cross section.I=e n S v……
4Solve for v, then find the time required for the average position to advance 1 m.v=Ie n S，t=Lv……
5Integrating the average velocity for alternating current gives the change in average position.x(t)=xm (1 − cos ωt)……
The following sections connect electron motion with electrical resistance and circuit laws.
6The relationship between voltage and electric current is derived.V=(mn e² τ)(LS)I……
7Resistivity is defined and used to calculate the mean time between collisions.R=ρ LS，ρ=mn e² τ……
8An empirical equation describes the change in resistivity with temperature.ρ≒ρ₀ (1 + α t)……
9The power delivered by the electric field gives the rate of Joule heating.P=V I=R I²……
10Conservation of electric charge and the properties of electric potential give 2 circuit laws.Σ I=0，Σ V=0……
11The terminal voltage is calculated by accounting for the battery's internal resistance.V=E−r I……
What this chapter has shown
The electrons' average position advances at speed v = I/(enS). For 1 A in a copper wire with a cross-sectional area of 1 mm², this speed is about 0.074 mm/s, and advancing 1 m takes about 3.8 hours. The large free-electron number density allows a large electric current even when the average position advances slowly.
For this alternating current, the average position moves back and forth over a range of 0.66 µm, returning to its starting point after each period.
In everyday lifeHousehold outlets supply alternating current. The electrons continue their thermal motion while their average position moves back and forth over a small distance. Using electricity does not require electrons to arrive from the power station.
Detailed summary
Electrons in a wire are repeatedly accelerated by the electric field and collide with atoms, resulting in a drift speed proportional to the electric field. Accounting for their motion gives Ohm's law, V = RI, and the energy transferred in collisions produces Joule heating. The electrons drift very slowly; the electric field propagates rapidly. In alternating current, the electrons move back and forth, remaining close to the same position.