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Wize University Physics Textbook (Master) > DC Circuits

Electric Current

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Electric Current




The electric current passing a point is the rate at which charge flows through it, i.e. the amount of charge per unit time:

 I=ΔQΔt \boxed{ \ I=\dfrac{\Delta{Q}}{\Delta{t}} \ }

  • QQ is the charge
  • tt is the time
  • II is the current in Amperes, 1 A=1 C/s1 \ A=1\ C/s

The smallest unit of charge is the elementary charge (the charge of an electron or proton). Since electric current is made up of an integer number of electrons, we have:

 Q=Ne \boxed{\ Q=Ne \ }

  • NN is the number of electrons
  • e=1.602×10−19e=1.602\times10^{-19} is the elementary charge

Practice: Electric Current


A current of 4.54.5 A passes through a circuit that is used to run a toaster.

a) If the current runs for 5.05.0 s, how much charge is transferred to the load?

b) How many electrons make up this charge?

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Microscopic Current Definition


Charges move in a medium with a velocity called the drift speed, and is given by:

 vd⃗ =a⃗τ=F⃗mτ=−eE⃗mτ \boxed{\ \vec{v_d}\ =\vec{a}\tau=\dfrac{\vec{F}}{m}\tau=\dfrac{-e\vec{E}}{m}\tau \ }

  • vd⃗\vec{v_d} is the drift speed of the electrons
  • τ\tau is the mean free time (time between collisions)
  • F⃗\vec{F} is the electric force
  • E⃗\vec{E} is the electric field
  • mm is the mass of the electron
  • ee is the elementary charge, e=1.602×10−19 Ce=1.602\times10^{-19} \ C


Wize Concept
The drift speed in any medium depends on:
  • the resistance of that medium (due to collisions between electrons and atoms which slows down the electrons)
  • the potential difference across the medium (which provides the electromotive force)

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The current density is the amount of current going through the cross section of a wire, and is given by:

 J⃗=I⃗A=−ne vd⃗ \boxed{\ \vec{J}=\frac{\vec{I}}{A}=-ne \ \vec{v_d}\ }
  • JJ is the current density
  • II is the current
  • AA is the cross-sectional area of the wire
  • nn is the electron density (number of electrons per unit volume)




Wize Concept
  • The electric field E⃗\vec{E} and current density J⃗\vec{J} always point in the same direction.
  • The drift speed of the electrons (negative charges) is in the opposite direction to the electric field and current density.


Wize Concept
Electric current is conserved, which means that the current will be the same at all points in the wire (similar to fluid flow!)


Exam Tip
Current density is inversely proportional to area, which means that:
  • the narrow parts have higher current density
  • the wider parts have lower current density

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Example: Drift Speed Comparison


The radius of Wire 1 is twice the radius of Wire 2 and they are made of the same material. If the current passing through Wire 1 is half the current passing through Wire 2, what is the drift speed of the electrons in Wire 2 compared to those in Wire 1?

Let's begin with the definition of current:

I=QtI=\dfrac{Q}{t}

The charge can be expressed in terms of the number of electrons as Q=NeQ=Ne. Putting this into the formula above we get:

=Net=\dfrac{Ne}{t}

Now using distance x=vdtx=v_dt, time can be written as t=xvdt=\dfrac{x}{v_d} and the formula becomes:

=Ne vdx=\dfrac{Ne\ v_d}{x}

Using volume V=AxV=Ax, the distance can be written as x=VAx=\dfrac{V}{A} and we have:

=NeA vdV=\dfrac{NeA\ v_d}{V}

Defining the electron number density as n=NVn=\dfrac{N}{V} we get:

=nAe vd= nAe\ v_d


Finally, the cross-sectional area is A=πr2A=\pi r^2 and we end up with:

I=πr2 ne vdI=\pi r^2\ ne\ v_d


Therefore the drift speed is given by:

vd=Iπr2 nev_d =\dfrac{I}{\pi r^2\ ne}


Write this equation for each of the wires:

For Wire 1 we have:

vd=I2π(2r)2 nev_d =\dfrac{\bcs I}{\bcs 2\pi (\bct{2r})^2\ ne}

For Wire 2 we have:
vd=Iπr2 nev_d =\dfrac{\bcs I}{\pi \bct{r}^2\ ne}

The is an extra factor of 18\dfrac{1}{8} in the formula for Wire 1, which means that v1=18 v2v_1=\dfrac{1}{8}\ v_2 , or v2=8v1v_2=8v_1.

Practice: Current Density and Drift Speed


An appliance is connected to a wire of diameter 2.562.56 mm , carrying a current of 6.06.0 A and a charge density of 8.48×10288.48\times10^{28} electrons/m3.

a) What is the current density?
b) What is the drift velocity?

Use e=1.602×10−19e=1.602\times10^{-19} C.