Fast or Slow?

A current in a wire arises from the motion (‘drift’) of electrons along it, under the action of an electric potential difference across its ends.

The current I in a wire of cross-sectional area a is related to the mean drift speed v of electrons by

I=avne

where n is the number of electrons per unit volume.

  1. Make sense of this equation.
  2. Use it to estimate the drift speed of electrons under the conditions:

    I=1A a=10-6m2 n=1027m-3 e=1.6×10-19C

  3. Compare this speed with the velocity of light.
  4. If you were told that magnetism comes from the fact that this electron motion has to be treated relativistically what would you say?

Solution

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Solution

  1. First let’s check the units.

    The LHS of this equation has units

    $$I \longrightarrow \frac{C}{s}$$

    The RHS has units

    $$Avne \longrightarrow m^2 \times ms^{-1} \times m^{-3} \times C = \frac{C}{s}$$

    So the units check out.

    Now let’s think about the meaning of the symbols.

    The current measures charge passing a point per second.

    It can be compared to the mass of water flowing down a pipe.

    It is then reasonable that I should increase with

    • the cross sectional area (think of increasing the cross section of a pipe)
    • the electron speed (think of making water go faster down a pipe)
    • the number of electrons in each unit volume (think of very dense water)
    • the charge carried by each electron (think of tweaking the mass of the water molecules)
  2. The data provided lead to

    $$v = \frac{I}{Ane} = \frac{1}{10^{-6} \times 10^{27} \times 1.6 \times 10^{-19}} \sim 10^{-2} ms^{-1}$$

  3. This is very small compared to c.
  4. We would therefore expect that it is quite safe to forget about relativistic effects.

    But we would be wrong!

    Magnetism does arise from relativistic effects associated with these small speeds.

    It comes from the fact that just as ‘moving clocks run slow’ so ‘moving lengths get contracted’. In this case the ‘moving length’ is the distance between moving charges.

    Since the speed involved is small, the contraction is tiny.

    There must be something LARGE to compensate.

    What is it?