The Centre of a Charged Hexagon

solved ElectrostaticPotential

Q. A regular hexagon $r_{ch} = r_{vn}$ of side 10 cm $r_{vn} = 10$ cm has a charge 5 μC $q_{v} = 5$ μC at each of its vertices $q_{h} = 6 \cdot q_{v}$ . Calculate the potential at the centre of the hexagon, in volts?

Inputs

side

r_{vn}

10 cm

charge

q_{v}

5 μC

Outputs

potential

V_{ch}

How it's solved

Solved with ElectrostaticPotential

Given

$r_{vn} = 10$ cm
$q_{v} = 5$ μC
$r_{ch} = r_{vn}$ from “A regular hexagon”
$q_{h} = 6 \cdot q_{v}$ from “at each of its vertices”

Find $V_{ch}$ .

Working

$r_{vn} = 10$ cm $= 0.1$ m
$q_{v} = 5$ μC $= 5 \times 10^{−6}$ C
at each of its vertices $q_{h} = 6 \cdot q_{v}$
$q_{h} = 6 \cdot 5 \times 10^{−6}$
$q_{h} = 3 \times 10^{−5}$ C
A regular hexagon $r_{ch} = r_{vn}$
$r_{ch} = 0.1$
$r_{ch} = 0.1$ m
potential of a point charge(centre, hexagon) $V_{ch} = \frac{k \cdot q_{h}}{r_{ch}}$
$V_{ch} = \frac{8.98755179 \times 10^{9} \cdot 3 \times 10^{−5}}{0.1}$
$V_{ch} = 2.696 \times 10^{6}$ V
$V_{ch} \approx 2.7 \times 10^{6}$ Vto 2 significant figures

Conclusion

A regular hexagon's vertices are each a side's length from its centre, so the six charges add their potentials there as if they were one charge six times as big.

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 :the_centre_of_a_charged_hexagon do4  kind :numerical5  describes "A regular hexagon of side %{side} has a charge %{charge} at each of its vertices. %{potential}, in " \6            "volts?"7  given side: "10 cm"8  given charge: "5 μC"9  asks potential: "Calculate the potential at the centre of the hexagon"10end

solution.rb

1require_relative "../questions/the_centre_of_a_charged_hexagon"2require_relative "../science/electrostatic_potential_and_capacitance/electrostatic_potential"34solve :the_centre_of_a_charged_hexagon, using: ElectrostaticPotential do5  objects :vertex, :neighbour, :hexagon, :centre67  take side: vertex.distance(from: neighbour)8  take charge: vertex.charge910  deduce_from "A regular hexagon", centre.distance(from: hexagon) => vertex.distance(from: neighbour)11  deduce_from "at each of its vertices", hexagon.charge => 6 * vertex.charge1213  potential = centre.potential(by: hexagon)14  answer potential:, in: "V", figures: 21516  conclusion "A regular hexagon's vertices are each a side's length from its centre, so the six charges add " \17             "their potentials there as if they were one charge six times as big."18end

playground.rb

1require_relative "../solutions/the_centre_of_a_charged_hexagon"23play_with :the_centre_of_a_charged_hexagon do4  title "The Centre of a Charged Hexagon"5end

The laws it drew on

quantity	written	units	within	from
coulomb constant	$k$	N m² C⁻²	—	ElectrostaticPotential
charge	$q$	C	—	ElectrostaticPotential
distance	$r$	m	—	ElectrostaticPotential
potential	$V$	V	—	ElectrostaticPotential
work to bring in	$W$	J	—	ElectrostaticPotential
potential of ground by vertex	$V_{gv}$	V	—
charge of vertex	$q_{v}$	C	—
distance of ground from vertex	$r_{gv}$	m	—
potential of ground by neighbour	$V_{gn}$	V	—
charge of neighbour	$q_{n}$	C	—
distance of ground from neighbour	$r_{gn}$	m	—
potential of ground by hexagon	$V_{gh}$	V	—
charge of hexagon	$q_{h}$	C	—
distance of ground from hexagon	$r_{gh}$	m	—
potential of ground by centre	$V_{gc}$	V	—
charge of centre	$q_{c}$	C	—
distance of ground from centre	$r_{gc}$	m	—
potential of vertex by ground	$V_{vg}$	V	—
charge of ground	$q_{g}$	C	—
distance of vertex from ground	$r_{v}$	m	—
potential of vertex by neighbour	$V_{vn}$	V	—
distance of vertex from neighbour	$r_{vn}$	m	—
potential of vertex by hexagon	$V_{vh}$	V	—
distance of vertex from hexagon	$r_{vh}$	m	—
potential of vertex by centre	$V_{vc}$	V	—
distance of vertex from centre	$r_{vc}$	m	—
potential of neighbour by ground	$V_{ng}$	V	—
distance of neighbour from ground	$r_{n}$	m	—
potential of neighbour by vertex	$V_{nv}$	V	—
distance of neighbour from vertex	$r_{nv}$	m	—
potential of neighbour by hexagon	$V_{nh}$	V	—
distance of neighbour from hexagon	$r_{nh}$	m	—
potential of neighbour by centre	$V_{nc}$	V	—
distance of neighbour from centre	$r_{nc}$	m	—
potential of hexagon by ground	$V_{hg}$	V	—
distance of hexagon from ground	$r_{h}$	m	—
potential of hexagon by vertex	$V_{hv}$	V	—
distance of hexagon from vertex	$r_{hv}$	m	—
potential of hexagon by neighbour	$V_{hn}$	V	—
distance of hexagon from neighbour	$r_{hn}$	m	—
potential of hexagon by centre	$V_{hc}$	V	—
distance of hexagon from centre	$r_{hc}$	m	—
potential of centre by ground	$V_{cg}$	V	—
distance of centre from ground	$r_{c}$	m	—
potential of centre by vertex	$V_{cv}$	V	—
distance of centre from vertex	$r_{cv}$	m	—
potential of centre by neighbour	$V_{cn}$	V	—
distance of centre from neighbour	$r_{cn}$	m	—
potential of centre by hexagon	$V_{ch}$	V	—
distance of centre from hexagon	$r_{ch}$	m	—
work to bring in on ground by vertex	$W_{gv}$	J	—
work to bring in on ground by neighbour	$W_{gn}$	J	—
work to bring in on ground by hexagon	$W_{gh}$	J	—
work to bring in on ground by centre	$W_{gc}$	J	—
work to bring in on vertex by ground	$W_{vg}$	J	—
work to bring in on vertex by neighbour	$W_{vn}$	J	—
work to bring in on vertex by hexagon	$W_{vh}$	J	—
work to bring in on vertex by centre	$W_{vc}$	J	—
work to bring in on neighbour by ground	$W_{ng}$	J	—
work to bring in on neighbour by vertex	$W_{nv}$	J	—
work to bring in on neighbour by hexagon	$W_{nh}$	J	—
work to bring in on neighbour by centre	$W_{nc}$	J	—
work to bring in on hexagon by ground	$W_{hg}$	J	—
work to bring in on hexagon by vertex	$W_{hv}$	J	—
work to bring in on hexagon by neighbour	$W_{hn}$	J	—
work to bring in on hexagon by centre	$W_{hc}$	J	—
work to bring in on centre by ground	$W_{cg}$	J	—
work to bring in on centre by vertex	$W_{cv}$	J	—
work to bring in on centre by neighbour	$W_{cn}$	J	—
work to bring in on centre by hexagon	$W_{ch}$	J	—

V_{ps} = \frac{k \cdot q_{s}}{r_{ps}}

potential of a point charge

W_{bs} = q_{b} \cdot V_{bs}

work is charge times potential

V_{gv} = \frac{k \cdot q_{v}}{r_{gv}}

potential of a point charge(ground, vertex)

V_{gn} = \frac{k \cdot q_{n}}{r_{gn}}

potential of a point charge(ground, neighbour)

V_{gh} = \frac{k \cdot q_{h}}{r_{gh}}

potential of a point charge(ground, hexagon)

V_{gc} = \frac{k \cdot q_{c}}{r_{gc}}

potential of a point charge(ground, centre)

V_{vg} = \frac{k \cdot q_{g}}{r_{v}}

potential of a point charge(vertex, ground)

V_{vn} = \frac{k \cdot q_{n}}{r_{vn}}

potential of a point charge(vertex, neighbour)

V_{vh} = \frac{k \cdot q_{h}}{r_{vh}}

potential of a point charge(vertex, hexagon)

V_{vc} = \frac{k \cdot q_{c}}{r_{vc}}

potential of a point charge(vertex, centre)

V_{ng} = \frac{k \cdot q_{g}}{r_{n}}

potential of a point charge(neighbour, ground)

V_{nv} = \frac{k \cdot q_{v}}{r_{nv}}

potential of a point charge(neighbour, vertex)

V_{nh} = \frac{k \cdot q_{h}}{r_{nh}}

potential of a point charge(neighbour, hexagon)

V_{nc} = \frac{k \cdot q_{c}}{r_{nc}}

potential of a point charge(neighbour, centre)

V_{hg} = \frac{k \cdot q_{g}}{r_{h}}

potential of a point charge(hexagon, ground)

V_{hv} = \frac{k \cdot q_{v}}{r_{hv}}

potential of a point charge(hexagon, vertex)

V_{hn} = \frac{k \cdot q_{n}}{r_{hn}}

potential of a point charge(hexagon, neighbour)

V_{hc} = \frac{k \cdot q_{c}}{r_{hc}}

potential of a point charge(hexagon, centre)

V_{cg} = \frac{k \cdot q_{g}}{r_{c}}

potential of a point charge(centre, ground)

V_{cv} = \frac{k \cdot q_{v}}{r_{cv}}

potential of a point charge(centre, vertex)

V_{cn} = \frac{k \cdot q_{n}}{r_{cn}}

potential of a point charge(centre, neighbour)

V_{ch} = \frac{k \cdot q_{h}}{r_{ch}}

potential of a point charge(centre, hexagon)

W_{gv} = q_{g} \cdot V_{gv}

work is charge times potential(ground, vertex)

W_{gn} = q_{g} \cdot V_{gn}

work is charge times potential(ground, neighbour)

W_{gh} = q_{g} \cdot V_{gh}

work is charge times potential(ground, hexagon)

W_{gc} = q_{g} \cdot V_{gc}

work is charge times potential(ground, centre)

W_{vg} = q_{v} \cdot V_{vg}

work is charge times potential(vertex, ground)

W_{vn} = q_{v} \cdot V_{vn}

work is charge times potential(vertex, neighbour)

W_{vh} = q_{v} \cdot V_{vh}

work is charge times potential(vertex, hexagon)

W_{vc} = q_{v} \cdot V_{vc}

work is charge times potential(vertex, centre)

W_{ng} = q_{n} \cdot V_{ng}

work is charge times potential(neighbour, ground)

W_{nv} = q_{n} \cdot V_{nv}

work is charge times potential(neighbour, vertex)

W_{nh} = q_{n} \cdot V_{nh}

work is charge times potential(neighbour, hexagon)

W_{nc} = q_{n} \cdot V_{nc}

work is charge times potential(neighbour, centre)

W_{hg} = q_{h} \cdot V_{hg}

work is charge times potential(hexagon, ground)

W_{hv} = q_{h} \cdot V_{hv}

work is charge times potential(hexagon, vertex)

W_{hn} = q_{h} \cdot V_{hn}

work is charge times potential(hexagon, neighbour)

W_{hc} = q_{h} \cdot V_{hc}

work is charge times potential(hexagon, centre)

W_{cg} = q_{c} \cdot V_{cg}

work is charge times potential(centre, ground)

W_{cv} = q_{c} \cdot V_{cv}

work is charge times potential(centre, vertex)

W_{cn} = q_{c} \cdot V_{cn}

work is charge times potential(centre, neighbour)

W_{ch} = q_{c} \cdot V_{ch}

work is charge times potential(centre, hexagon)

r_{ch} = r_{vn}

A regular hexagon

q_{h} = 6 \cdot q_{v}

at each of its vertices