Two Small Spheres Thirty Centimetres Apart

solved CoulombsLaw

Q. Two small charged spheres having charges of 2 × 10⁻⁷ C $q_{f} = 2 \times 10^{−7}$ C and 3 × 10⁻⁷ C $q_{s} = 3 \times 10^{−7}$ C are placed 30 cm $r_{fs} = 30$ cm apart in air. What is the force between them, in newtons?

Inputs

first

q_{f}

2e-07 C

second

q_{s}

3e-07 C

apart

r_{fs}

30 cm

Outputs

force

F_{e,fs}

How it's solved

Solved with CoulombsLaw

Given

$q_{f} = 2 \times 10^{−7}$ C
$q_{s} = 3 \times 10^{−7}$ C
$r_{fs} = 30$ cm

Find $F_{e,fs}$ .

Working

$r_{fs} = 30$ cm $= 0.3$ m
coulombs law(first, second) $F_{e,fs} = \frac{k \cdot q_{f} \cdot q_{s}}{{r_{fs}}^{2}}$
$F_{e,fs} = \frac{8.98755179 \times 10^{9} \cdot 2 \times 10^{−7} \cdot 3 \times 10^{−7}}{{0.3}^{2}}$
$F_{e,fs} = 0.005992$ N
$F_{e,fs} \approx 0.0060$ Nto 2 significant figures

Conclusion

Like charges push each other apart with a force that grows with each charge and falls off as the square of the distance between them.

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 :two_small_spheres_thirty_centimetres_apart do4  kind :numerical5  describes "Two small charged spheres having charges of %{first} and %{second} are placed %{apart} apart in air. " \6            "%{force}, in newtons?"7  given first: "2e-7 C"8  given second: "3e-7 C"9  given apart: "30 cm"10  asks force: "What is the force between them"11end

solution.rb

1require_relative "../questions/two_small_spheres_thirty_centimetres_apart"2require_relative "../science/electric_charges_and_fields/coulombs_law"34solve :two_small_spheres_thirty_centimetres_apart, using: CoulombsLaw do5  objects :first, :second67  take first: first.charge8  take second: second.charge9  take apart: first.separation(from: second)1011  force = first.electric_force(by: second)12  answer force:, in: "N", figures: 21314  conclusion "Like charges push each other apart with a force that grows with each charge and falls off as the " \15             "square of the distance between them."16end

playground.rb

1require_relative "../solutions/two_small_spheres_thirty_centimetres_apart"23play_with :two_small_spheres_thirty_centimetres_apart do4  title "Two Small Spheres Thirty Centimetres Apart"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)