Electrons Drifting in Copper

solved DriftVelocity

Q. A copper wire of cross-sectional area 1.0 × 10⁻⁷ m² $A = 1.0 \times 10^{−7}$ m² carries a current of 1.5 A $I = 1.5$ A. Assume that each copper atom contributes roughly one conduction electron. The density of copper is 9.0 × 10³ kg/m³ $ρ = 9.0 \times 10^{3}$ kg/m³, and its atomic mass is 63.5 g/mol $M = 63.5$ g/mol. What is the average drift speed of the conduction electrons, in mm/s?

Inputs

area

A

1e-07 m²

current

I

1.5 A

density

ρ

9000 kg/m³

molar mass

M

63.5 g/mol

Outputs

drift

v_{d}

How it's solved

Solved with DriftVelocity

Given

$A = 1.0 \times 10^{−7}$ m²
$I = 1.5$ A
$ρ = 9.0 \times 10^{3}$ kg/m³
$M = 63.5$ g/mol

Find $v_{d}$ .

Working

$M = 63.5$ g/mol $= 0.0635$ kg/mol
one free electron an atom(wire) $n = \frac{ρ \cdot N_{A}}{M}$
$n = \frac{9000 \cdot 6.02214076 \times 10^{23}}{0.0635}$
$n = 8.53531761 \times 10^{28}$ m⁻³
current from drift(wire) $I = n \cdot e \cdot A \cdot v_{d}$
$1.5 = 8.53531761 \times 10^{28} \cdot 1.60217663 \times 10^{−19} \cdot 1 \times 10^{−7} \cdot v_{d}$
$v_{d} = 1.097$ mm/s
$v_{d} = 0.00109688521$ m/s $= 1.09688521$ mm/s
$v_{d} \approx 1.1$ mm/sto 2 significant figures

Conclusion

Copper holds so many free electrons that a current of amperes needs them to drift only about a millimetre a second.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :electrons_drifting_in_copper do4  kind :numerical5  describes "A copper wire of cross-sectional area %{area} carries a current of %{current}. Assume that each copper " \6            "atom contributes roughly one conduction electron. The density of copper is %{density}, and its atomic " \7            "mass is %{molar_mass}. %{drift}, in mm/s?"8  given area: "1.0e-7 m2"9  given current: "1.5 A"10  given density: "9.0e3 kg/m3"11  given molar_mass: "63.5 g/mol"12  asks drift: "What is the average drift speed of the conduction electrons"13end

solution.rb

1require_relative "../questions/electrons_drifting_in_copper"2require_relative "../science/current_electricity/drift_velocity"34solve :electrons_drifting_in_copper, using: DriftVelocity do5  objects :wire67  take area: wire.cross_section8  take current: wire.current9  take density: wire.density10  take molar_mass: wire.molar_mass1112  drift = wire.drift_speed13  answer drift:, in: "mm/s", figures: 21415  conclusion "Copper holds so many free electrons that a current of amperes needs them to drift only about " \16             "a millimetre a second."17end

playground.rb

1require_relative "../solutions/electrons_drifting_in_copper"23play_with :electrons_drifting_in_copper do4  title "Electrons Drifting in Copper"5end

The laws it drew on

quantity	written	units	within	from
elementary charge	$e$	C	—	DriftVelocity
avogadro constant	$N_{A}$	mol⁻¹	—	DriftVelocity
current	$I$	A	—	DriftVelocity
cross section	$A$	m²	—	DriftVelocity
density	$ρ$	kg/m³	0 … ∞	DriftVelocity
molar mass	$M$	kg/mol	—	DriftVelocity
free electrons	$n$	m⁻³	—	DriftVelocity
drift speed	$v_{d}$	m/s	—	DriftVelocity
length of the conductor	$l$	m	—	DriftVelocity
drift time	$t$	s	—	DriftVelocity
free electrons of ground	$n_{g}$	m⁻³	—
density of ground	$ρ_{g}$	kg/m³	0 … ∞
molar mass of ground	$M_{g}$	kg/mol	—
free electrons of wire	$n$	m⁻³	—
density of wire	$ρ$	kg/m³	0 … ∞
molar mass of wire	$M$	kg/mol	—
current of ground	$I_{g}$	A	—
cross section of ground	$A_{g}$	m²	—
drift speed of ground	$v_{d,g}$	m/s	—
current of wire	$I$	A	—
cross section of wire	$A$	m²	—
drift speed of wire	$v_{d}$	m/s	—
length of the conductor of ground	$l_{g}$	m	—
drift time of ground	$t_{g}$	s	—
length of the conductor of wire	$l$	m	—
drift time of wire	$t$	s	—

n_{w} = \frac{ρ_{w} \cdot N_{A}}{M_{w}}

one free electron an atom

I_{w} = n_{w} \cdot e \cdot A_{w} \cdot v_{d,w}

current from drift

l_{w} = v_{d,w} \cdot t_{w}

drifting the length of the wire

n_{g} = \frac{ρ_{g} \cdot N_{A}}{M_{g}}

one free electron an atom(ground)

n = \frac{ρ \cdot N_{A}}{M}

one free electron an atom(wire)

I_{g} = n_{g} \cdot e \cdot A_{g} \cdot v_{d,g}

current from drift(ground)

I = n \cdot e \cdot A \cdot v_{d}

current from drift(wire)

l_{g} = v_{d,g} \cdot t_{g}

drifting the length of the wire(ground)

l = v_{d} \cdot t

drifting the length of the wire(wire)