An Electron Pulled by a Proton

Q. An electron and a proton are 1 × 10⁻¹⁰ m apart. Taking the charge of an electron as -1.6 × 10⁻¹⁹ C and its mass as 9.11 × 10⁻³¹ kg, what is the electron's acceleration away from the proton, due to the electrical force between them, in m/s²? expects -2.5 × 10²² m/s²

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The web of concepts

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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?

The playground

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}

The solution

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 Ruby

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

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

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

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)