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Electric current Drift velocity
Drift velocity and current
Interactive graphic for drift velocity
Picture 2.6 All the electrons in a cylinder with volume, V = A v t flow past a point in a time t.

How fast must free electrons move in a wire to produce a decent current?

‘Current’ means the rate at which electric charge flows past a point in a circuit. Imagine standing at point X with a stopwatch and timing the charge flowing past. (We have to imagine that all the electrons move at the same speed, v.) We'll watch what happens to the electron highlighted in red.

Suppose you start your watch and let it run for a time, t. The highlighted electron electron will have travelled a distance . In fact, in time t, all of the electrons in the cylinder of length have flowed past you.

So what current has flowed? We need to work out how much charge has passed point A.
We start by thinking of the volume of the cylinder.

Volume of cylinder = A × where A is the cross-sectional area of the wire
If concentration of electrons in the metal is n per cubic metre then:
Number of electrons in cylinder = n × A ×
             If each electron carries charge Q then:
Charge carried by electrons in cylinder = n × A × × Q
             But the length of the cylinder is v * t   where v is the drift velocity and t is the time we used
             So:
Charge carried by electrons in cylinder = n × A × v × t × Q
This is the amount of charge which passes point A in time t. To find the current which this represents, we need to find the rate at which the charge has flowed. So we divide by the time t.
Current = charge / time = n × A × v × t × Q / t    =   n A v Q
So the electric current I flowing in a wire is given by
I = n a v q where n is the number of electrons per cubic metre
A is the cross sectional area of the wire
v is the drift velocity of the electrons
Q is the charge of an electron

Resistivity and charge density
A material with a lot of free electrons (a high value of n) can carry a current more easily than one with a smaller charge density. To carry a given current, the electrons don't have to move very fast because there are so many of them to carry the charge This means that they rarely collide with atoms or impurities in the metal, and so it is a good conductor.

Semiconductors are materials with few free electrons – perhaps one-millionth of copper’s concentration. So free electrons in semiconductors have to have much higher drift velocities to carry the same current. Their speed has to make up for the smaller amount of charge that is moving. Therefore they collide with atoms much more often. The resistivity of a semiconductor is typically one million times that of copper.

How fast do electrons drift?
We can get an idea of how fast the drift velocity is by taking some typical values of current and wire dimensions.

Let's think of a current of 5 A that is flowing in a copper wire with a cross section of 0.5 mm2 (= 0.5 * 10-6 m2)

For copper, n = 8.5 × 1028 per m3
The charge on an electron, Q = 1.6 × 10-19 C

So,
   I = n A v Q

   5 = 8.5 × 1028 × 0.5 × 10-6 × v × 1.6 × 10-19

   5 = 27 200 v

   v = 7.35 × 10-4 m s-1

So, for this current, the drift velocity of electrons is about a tenth of a millimetre per second: pretty slow!

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Question 5
a) The equation for current I = nAQv can be rearranged to give an equation for drift velocity:
v = I / nAQ
Use this equation to decide which of the following statements are true. In each case, give a brief explanation.
True False
i. The drift velocity of electrons in a wire is proportional to the current that flows in the wire.
ii. For a material such as copper, which has a high concentration of free electrons, the drift velocity is less than for a material with a lower concentration.
iii. If a current flows from a thick copper wire into a thinner copper wire, the electrons will slow down (their drift velocity decreases).

b) Calculate the drift velocity of electrons in a copper wire of cross-sectional area 1 mm2 when a current of 2.5 A flows through it.

For copper, n = 8.5 × 1028 per m3
The charge on an electron, Q = 1.6 × 10-19 C