How Far Apart the Spheres Are

solved CoulombsLaw

Q. The electrostatic force on a small sphere of charge 0.4 μC $q_{f} = 0.4$ μC due to another small sphere of charge –0.8 μC $q_{s} = 0.8$ μC in air is 0.2 N $F_{e,fs} = 0.2$ N. What is the distance between the two spheres, in cm?

Inputs

first

q_{f}

0.4 μC

second

q_{s}

0.8 μC

force

F_{e,fs}

0.2 N

Outputs

distance

r_{fs}

How it's solved

Solved with CoulombsLaw

Given

$q_{f} = 0.4$ μC
$q_{s} = 0.8$ μC
$F_{e,fs} = 0.2$ N

Find $r_{fs}$ .

Working

$q_{f} = 0.4$ μC $= 4 \times 10^{−7}$ C
$q_{s} = 0.8$ μC $= 8 \times 10^{−7}$ C
coulombs law(first, second) $F_{e,fs} = \frac{k \cdot q_{f} \cdot q_{s}}{{r_{fs}}^{2}}$
$0.2 = \frac{8.98755179 \times 10^{9} \cdot 4 \times 10^{−7} \cdot 8 \times 10^{−7}}{{r_{fs}}^{2}}$
$r_{fs} = 11.99$ cm
$r_{fs} = 0.119916983$ m $= 11.9916983$ cm
$r_{fs} \approx 12$ cmto 2 significant figures

Conclusion

The force falls off as the square of the distance, so knowing the charges and the pull between them fixes how far apart they must be.

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 :how_far_apart_the_spheres_are do4  kind :numerical5  describes "The electrostatic force on a small sphere of charge %{first} due to another small sphere of charge " \6            "–%{second} in air is %{force}. %{distance}, in cm?"7  given first: "0.4 μC"8  given second: "0.8 μC"9  given force: "0.2 N"10  asks distance: "What is the distance between the two spheres"11end

solution.rb

1require_relative "../questions/how_far_apart_the_spheres_are"2require_relative "../science/electric_charges_and_fields/coulombs_law"34solve :how_far_apart_the_spheres_are, using: CoulombsLaw do5  objects :first, :second67  take first: first.charge8  take second: second.charge9  take force: first.electric_force(by: second)1011  distance = first.separation(from: second)12  answer distance:, in: "cm", figures: 21314  conclusion "The force falls off as the square of the distance, so knowing the charges and the pull between " \15             "them fixes how far apart they must be."16end

playground.rb

1require_relative "../solutions/how_far_apart_the_spheres_are"23play_with :how_far_apart_the_spheres_are do4  title "How Far Apart the Spheres Are"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
electric force on ground by first	$F_{e,gf}$	N	—
charge of ground	$q_{g}$	C	—
charge of first	$q_{f}$	C	—
separation of ground from first	$r_{gf}$	m	0 … ∞
electric force on ground by second	$F_{e,gs}$	N	—
charge of second	$q_{s}$	C	—
separation of ground from second	$r_{gs}$	m	0 … ∞
electric force on first by ground	$F_{e,fg}$	N	—
separation of first from ground	$r_{f}$	m	0 … ∞
electric force on first by second	$F_{e,fs}$	N	—
separation of first from second	$r_{fs}$	m	0 … ∞
electric force on second by ground	$F_{e,sg}$	N	—
separation of second from ground	$r_{s}$	m	0 … ∞
electric force on second by first	$F_{e,sf}$	N	—
separation of second from first	$r_{sf}$	m	0 … ∞

F_{e,oo} = \frac{k \cdot q_{o} \cdot q_{o}}{{r_{oo}}^{2}}

coulombs law

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

coulombs law(ground, first)

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

coulombs law(ground, second)

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

coulombs law(first, ground)

F_{e,fs} = \frac{k \cdot q_{f} \cdot q_{s}}{{r_{fs}}^{2}}

coulombs law(first, second)

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

coulombs law(second, ground)

F_{e,sf} = \frac{k \cdot q_{s} \cdot q_{f}}{{r_{sf}}^{2}}

coulombs law(second, first)