An Electron Pulled by a Proton

solved CoulombsLaw · NewtonsSecondLaw

Q. An electron and a proton $q_{p} = - q_{e}$ are 1 × 10⁻¹⁰ m $r_{ep} = 1 \times 10^{−10}$ m apart. Taking the charge of an electron as -1.6 × 10⁻¹⁹ C $q_{e} = −1.6 \times 10^{−19}$ C and its mass as 9.11 × 10⁻³¹ kg $m_{e} = 9.11 \times 10^{−31}$ kg, what is the electron's acceleration away from the proton, due to the electrical force between them $F_{net,e} = F_{e,ep}$ , in m/s²?

Inputs

separation

r_{ep}

0.00 m

electron

q_{e}

-1.6e-19 C

mass

m_{e}

9.11e-31 kg

Outputs

acceleration

a_{e}

How it's solved

Solved with CoulombsLaw, NewtonsSecondLaw

Given

$r_{ep} = 1 \times 10^{−10}$ m
$q_{e} = −1.6 \times 10^{−19}$ C
$m_{e} = 9.11 \times 10^{−31}$ kg
$q_{p} = - q_{e}$ from “a proton”
$F_{net,e} = F_{e,ep}$ from “due to the electrical force between them”

Find $a_{e}$ .

Working

a proton $q_{p} = - q_{e}$
$q_{p} = - (−1.6 \times 10^{−19})$
$q_{p} = 1.6 \times 10^{−19}$ C
coulombs law(electron, proton) $F_{e,ep} = \frac{k \cdot q_{e} \cdot q_{p}}{{r_{ep}}^{2}}$
$F_{e,ep} = \frac{8.98755179 \times 10^{9} \cdot (−1.6 \times 10^{−19}) \cdot 1.6 \times 10^{−19}}{{(1 \times 10^{−10})}^{2}}$
$F_{e,ep} = −2.30081326 \times 10^{−8}$ N
due to the electrical force between them $F_{net,e} = F_{e,ep}$
$F_{net,e} = (−2.30081326 \times 10^{−8})$
$F_{net,e} = −2.30081326 \times 10^{−8}$ N
second law(pulled) $F_{net,e} = m_{e} \cdot a_{e}$
$(−2.30081326 \times 10^{−8}) = 9.11 \times 10^{−31} \cdot a_{e}$
$a_{e} = −2.526 \times 10^{22}$ m/s²
$a_{e} \approx −2.5 \times 10^{22}$ m/s²to 2 significant figures

Conclusion

The pull between them is tiny by everyday standards, but the electron is so light that it accelerates enormously; the minus sign says the acceleration is towards the proton.

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 :an_electron_pulled_by_a_proton do4  kind :numerical5  describes "An electron and a proton are %{separation} apart. Taking the charge of an electron as %{electron} and " \6            "its mass as %{mass}, %{acceleration}, in m/s²?"7  given separation: "1e-10 m"8  given electron: "-1.6e-19 C"9  given mass: "9.11e-31 kg"10  asks acceleration: "what is the electron's acceleration away from the proton, due to the electrical force between them"11end

solution.rb

1require_relative "../questions/an_electron_pulled_by_a_proton"2require_relative "../science/electric_charges_and_fields/coulombs_law"3require_relative "../science/laws_of_motion/newtons_second_law"45solve :an_electron_pulled_by_a_proton, using: [ CoulombsLaw, NewtonsSecondLaw ] do6  objects :electron, :proton7  electron.goes(:pulled)89  take separation: electron.separation(from: proton)10  take electron: electron.charge11  take mass: electron.mass1213  deduce_from "a proton", proton.charge => -electron.charge14  deduce_from "due to the electrical force between them", electron.net_force => electron.electric_force(by: proton)1516  take acceleration: electron.acceleration17  answer find_value(:acceleration, in: "m/s2", figures: 2)1819  conclusion "The pull between them is tiny by everyday standards, but the electron is so light that it " \20             "accelerates enormously; the minus sign says the acceleration is towards the proton."21end

playground.rb

1require_relative "../solutions/an_electron_pulled_by_a_proton"23play_with :an_electron_pulled_by_a_proton do4  title "An Electron Pulled by a Proton"5end

The laws it drew on

quantity	written	units	within	from
charge	$q$	C	—	CoulombsLaw
coulomb constant	$k$	N m² C⁻²	—	CoulombsLaw
separation	$r$	m	0 … ∞	CoulombsLaw
electric force	$F_{e}$	N	—	CoulombsLaw
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 pulled	$Δ x_{e}$	m	—
position of electron at end of pulled	$x_{1,e}$	m	—
position of electron at start of pulled	$x_{0,e}$	m	—
average velocity of electron during pulled	$v_{av,e}$	m/s	—
time taken of pulled	$Δ t$	s	0 … 1e+05
average speed of electron during pulled	$c_{av,e}$	m/s	—
path length of electron during pulled	$d_{e}$	m	—
change of velocity of electron during pulled	$Δ v_{e}$	m/s	—
velocity of electron at end of pulled from ground	$v_{1,e}$	m/s	—
velocity of electron at start of pulled from ground	$v_{0,e}$	m/s	—
average acceleration of electron during pulled	$a_{av,e}$	m/s²	—
acceleration of electron during pulled	$a_{e}$	m/s²	—
net force on electron during pulled	$F_{net,e}$	N	—
mass of electron	$m_{e}$	kg	0 … ∞
electric force on ground by electron	$F_{e,ge}$	N	—
charge of ground	$q_{g}$	C	—
charge of electron	$q_{e}$	C	—
separation of ground from electron	$r_{ge}$	m	0 … ∞
electric force on ground by proton	$F_{e,gp}$	N	—
charge of proton	$q_{p}$	C	—
separation of ground from proton	$r_{gp}$	m	0 … ∞
electric force on electron by ground	$F_{e,eg}$	N	—
separation of electron from ground	$r_{e}$	m	0 … ∞
electric force on electron by proton	$F_{e,ep}$	N	—
separation of electron from proton	$r_{ep}$	m	0 … ∞
electric force on proton by ground	$F_{e,pg}$	N	—
separation of proton from ground	$r_{p}$	m	0 … ∞
electric force on proton by electron	$F_{e,pe}$	N	—
separation of proton from electron	$r_{pe}$	m	0 … ∞
mass of proton	$m_{p}$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on electron	$W_{e}$	N	—
weight on proton	$W_{p}$	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_{e,oo} = \frac{k \cdot q_{o} \cdot q_{o}}{{r_{oo}}^{2}}

coulombs law

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_{e} = x_{1,e} - x_{0,e}

change of position(pulled)

v_{av,e} = \frac{Δ x_{e}}{Δ t}

displacement a time(pulled)

c_{av,e} = \frac{d_{e}}{Δ t}

path a time(pulled)

Δ v_{e} = v_{1,e} - v_{0,e}

velocity gained(pulled)

a_{av,e} = \frac{Δ v_{e}}{Δ t}

velocity gained a time(pulled)

v_{1,e} = v_{0,e} + a_{e} \cdot Δ t

velocity after a time(pulled)

x_{1,e} = x_{0,e} + v_{0,e} \cdot Δ t + \frac{a_{e} \cdot {Δ t}^{2}}{2}

position after a time(pulled)

{v_{1,e}}^{2} = {v_{0,e}}^{2} + 2 \cdot a_{e} \cdot (x_{1,e} - x_{0,e})

velocity after a distance(pulled)

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

mean of the velocities(pulled)

F_{net,e} = m_{e} \cdot a_{e}

second law(pulled)

F_{e,ge} = \frac{k \cdot q_{g} \cdot q_{e}}{{r_{ge}}^{2}}

coulombs law(ground, electron)

F_{e,gp} = \frac{k \cdot q_{g} \cdot q_{p}}{{r_{gp}}^{2}}

coulombs law(ground, proton)

F_{e,eg} = \frac{k \cdot q_{e} \cdot q_{g}}{{r_{e}}^{2}}

coulombs law(electron, ground)

F_{e,ep} = \frac{k \cdot q_{e} \cdot q_{p}}{{r_{ep}}^{2}}

coulombs law(electron, proton)

F_{e,pg} = \frac{k \cdot q_{p} \cdot q_{g}}{{r_{p}}^{2}}

coulombs law(proton, ground)

F_{e,pe} = \frac{k \cdot q_{p} \cdot q_{e}}{{r_{pe}}^{2}}

coulombs law(proton, electron)

q_{p} = - q_{e}

a proton

F_{net,e} = F_{e,ep}

due to the electrical force between them

m_{e} \neq 0

has mass?(electron)

m_{p} \neq 0

has mass?(proton)

F_{net,e} = 0

free of forces?(electron, pulled)