The Centre of a Charged Hexagon

Q. A regular hexagon of side 10 cm has a charge 5 μC at each of its vertices. Calculate the potential at the centre of the hexagon, in volts? expects 2.7 × 10⁶ V

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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?

The playground

Inputs

side

r_{vn}

10 cm

charge

q_{v}

5 μC

Outputs

potential

V_{ch}

The solution

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 Ruby

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

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

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

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