An Electron Falls in a Field

Q. An electron falls from rest through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 10⁴ N/C. Taking its charge as 1.6 × 10⁻¹⁹ C and its mass as 9.11 × 10⁻³¹ kg, what is the time of fall, in seconds? expects 2.9 × 10⁻⁹ s

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Q. An electron falls from rest $v_{0} = 0$ m/s, $x_{0} = 0$ m through a distance of 1.5 cm $x_{1} = 1.5$ cm in a uniform electric field $F_{net} = F$ of magnitude 2.0 × 10⁴ N/C $E = 2.0 \times 10^{4}$ N/C. Taking its charge as 1.6 × 10⁻¹⁹ C $q = 1.6 \times 10^{−19}$ C and its mass as 9.11 × 10⁻³¹ kg $m = 9.11 \times 10^{−31}$ kg, what is the time of fall, in seconds?

The playground

Inputs

distance

x_{1}

1.5 cm

field

E

2e+04 N/C

charge

q

1.6e-19 C

mass

m

9.11e-31 kg

Outputs

time

Δ t

The solution

Solved with ElectricField, NewtonsSecondLaw

Given

$E = 2.0 \times 10^{4}$ N/C
$q = 1.6 \times 10^{−19}$ C
$m = 9.11 \times 10^{−31}$ kg
$x_{1} = 1.5$ cm
$v_{0} = 0$ m/s from “falls from rest”
$x_{0} = 0$ m from “falls from rest”
$F_{net} = F$ from “in a uniform electric field”

Find $Δ t$ .

Working

$E = 2.0 \times 10^{4}$ N/C $= 20000$ V/m
$x_{1} = 1.5$ cm $= 0.015$ m
force on a charge in a field(electron) $F = q \cdot E$
$F = 1.6 \times 10^{−19} \cdot 20000$
$F = 3.2 \times 10^{−15}$ N
in a uniform electric field $F_{net} = F$
$F_{net} = 3.2 \times 10^{−15}$
$F_{net} = 3.2 \times 10^{−15}$ N
second law(falling) $F_{net} = m \cdot a$
$3.2 \times 10^{−15} = 9.11 \times 10^{−31} \cdot a$
$a = 3.51262349 \times 10^{15}$ m/s²
position after a time(falling) $x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}$
$0.015 = 0 + 0 \cdot Δ t + \frac{3.51262349 \times 10^{15} \cdot {Δ t}^{2}}{2}$
$Δ t = 2.922 \times 10^{−9}$ s
$Δ t \approx 2.9 \times 10^{−9}$ sto 2 significant figures

Conclusion

Unlike a fall under gravity, the field's pull grows with the charge but not with the mass, so the light electron is flung through the gap in nanoseconds.

The Ruby

1require_relative "../physics"23question :an_electron_falls_in_a_field do4  kind :numerical5  describes "An electron falls from rest through a distance of %{distance} in a uniform electric field of magnitude " \6            "%{field}. Taking its charge as %{charge} and its mass as %{mass}, %{time}, in seconds?"7  given distance: "1.5 cm"8  given field: "2.0e4 N/C"9  given charge: "1.6e-19 C"10  given mass: "9.11e-31 kg"11  asks time: "what is the time of fall"12end

1require_relative "../questions/an_electron_falls_in_a_field"2require_relative "../science/electric_charges_and_fields/electric_field"3require_relative "../science/laws_of_motion/newtons_second_law"45solve :an_electron_falls_in_a_field, using: [ ElectricField, NewtonsSecondLaw ] do6  objects :electron7  electron.goes(:falling)89  take field: E10  take charge: electron.charge11  take mass: electron.mass12  take distance: electron.final_position1314  deduce_from "falls from rest", electron.initial_velocity => "0 m/s", electron.initial_position => "0 m"15  deduce_from "in a uniform electric field", electron.net_force => electron.force_of_the_field1617  take time: falling.time_taken18  answer find_value(:time, in: "s", figures: 2)1920  conclusion "Unlike a fall under gravity, the field's pull grows with the charge but not with the mass, so " \21             "the light electron is flung through the gap in nanoseconds."22end

1require_relative "../solutions/an_electron_falls_in_a_field"23play_with :an_electron_falls_in_a_field do4  title "An Electron Falls in a Field"5end

The laws

quantity	written	units	within	from
charge	$q$	C	—	ElectricField
coulomb constant	$k$	N m² C⁻²	—	ElectricField
field strength	$E$	V/m	—	ElectricField
separation	$r$	m	0 … ∞	ElectricField
field of a charge	$E_{q}$	V/m	—	ElectricField
force of the field	$F$	N	—	ElectricField
volume	$volume$	—	—	Vectors
mass	$m$	kg	0 … ∞	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$t$	s	0 … 1e+05	Vectors
power	$power$	—	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
acceleration	$a$	m/s²	—	Vectors
velocity	$v$	m/s	—	Vectors
displacement	$Δ x$	m	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$F$	N	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	$v_{av}$	m/s	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors
position	$x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
gravity	$g$	m/s²	—	UniformAcceleration
net force	$F_{net}$	N	—	NewtonsSecondLaw
weight	$W$	N	—	NewtonsSecondLaw
displacement of electron during falling	$Δ x$	m	—
position of electron at end of falling	$x_{1}$	m	—
position of electron at start of falling	$x_{0}$	m	—
average velocity of electron during falling	$v_{av}$	m/s	—
time taken of falling	$Δ t$	s	0 … 1e+05
average speed of electron during falling	$c_{av}$	m/s	—
path length of electron during falling	$d$	m	—
change of velocity of electron during falling	$Δ v$	m/s	—
velocity of electron at end of falling from ground	$v_{1}$	m/s	—
velocity of electron at start of falling from ground	$v_{0}$	m/s	—
average acceleration of electron during falling	$a_{av}$	m/s²	—
acceleration of electron during falling	$a$	m/s²	—
net force on electron during falling	$F_{net}$	N	—
mass of electron	$m$	kg	0 … ∞
force of the field on ground	$F_{g}$	N	—
charge of ground	$q_{g}$	C	—
force of the field on electron	$F$	N	—
charge of electron	$q$	C	—
field of a charge of ground by electron	$E_{q,g}$	V/m	—
separation of ground from electron	$r_{g}$	m	0 … ∞
field of a charge of electron by ground	$E_{q,g}$	V/m	—
separation of electron from ground	$r$	m	0 … ∞
weight on ground	$W_{g}$	N	—
weight on electron	$W$	N	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

v_{0} \equiv v_{i} (start of motion)

starting velocity

v \equiv v_{i} (end of motion)

final velocity

t \equiv Δ t_{m}

time

F_{b} = q_{b} \cdot E

force on a charge in a field

E_{q,ps} = \frac{k \cdot q_{s}}{{r_{ps}}^{2}}

field of a point charge

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

a_{i} (part 1) = a_{i} (whole)

steady through its parts

m_{b} \neq 0

has mass?

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y

Δ x = x_{1} - x_{0}

change of position(falling)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(falling)

c_{av} = \frac{d}{Δ t}

path a time(falling)

Δ v = v_{1} - v_{0}

velocity gained(falling)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(falling)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(falling)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(falling)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(falling)

x_{1} - x_{0} = \frac{(v_{1} + v_{0}) \cdot Δ t}{2}

mean of the velocities(falling)

F_{net} = m \cdot a

second law(falling)

F_{g} = q_{g} \cdot E

force on a charge in a field(ground)

F = q \cdot E

force on a charge in a field(electron)

E_{q,g} = \frac{k \cdot q}{{r_{g}}^{2}}

field of a point charge(ground, electron)

E_{q,g} = \frac{k \cdot q_{g}}{r^{2}}

field of a point charge(electron, ground)

F_{net} = F

in a uniform electric field

m \neq 0

has mass?(electron)

F_{net} = 0

free of forces?(electron, falling)