Bringing In a Charge

solved ElectrostaticPotential

Q. A charge of 4 × 10⁻⁷ C $q_{s} = 4 \times 10^{−7}$ C is located 9 cm $r_{bs} = 9$ cm from a point P. What is the work done in bringing a charge of 2 × 10⁻⁹ C $q_{b} = 2 \times 10^{−9}$ C from infinity to the point P, in joules?

Inputs

source

q_{s}

4e-07 C

distance

r_{bs}

9 cm

brought

q_{b}

2e-09 C

Outputs

work

W_{bs}

How it's solved

Solved with ElectrostaticPotential

Given

$q_{s} = 4 \times 10^{−7}$ C
$r_{bs} = 9$ cm
$q_{b} = 2 \times 10^{−9}$ C

Find $W_{bs}$ .

Working

$r_{bs} = 9$ cm $= 0.09$ m
potential of a point charge(brought, source) $V_{bs} = \frac{k \cdot q_{s}}{r_{bs}}$
$V_{bs} = \frac{8.98755179 \times 10^{9} \cdot 4 \times 10^{−7}}{0.09}$
$V_{bs} = 39944.6746$ V
work is charge times potential(brought, source) $W_{bs} = q_{b} \cdot V_{bs}$
$W_{bs} = 2 \times 10^{−9} \cdot 39944.6746$
$W_{bs} = 7.989 \times 10^{−5}$ J
$W_{bs} \approx 8.0 \times 10^{−5}$ Jto 2 significant figures

Conclusion

The potential at P is the work per unit charge to bring a charge there from far away, by any path; the charge brought in times that potential is the work.

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 :bringing_in_a_charge do4  kind :numerical5  describes "A charge of %{source} is located %{distance} from a point P. %{work} in bringing a charge of %{brought} " \6            "from infinity to the point P, in joules?"7  given source: "4e-7 C"8  given distance: "9 cm"9  given brought: "2e-9 C"10  asks work: "What is the work done"11end

solution.rb

1require_relative "../questions/bringing_in_a_charge"2require_relative "../science/electrostatic_potential_and_capacitance/electrostatic_potential"34solve :bringing_in_a_charge, using: ElectrostaticPotential do5  objects :source, :brought67  take source: source.charge8  take distance: brought.distance(from: source)9  take brought: brought.charge1011  work = brought.work_to_bring_in(by: source)12  answer work:, in: "J", figures: 21314  conclusion "The potential at P is the work per unit charge to bring a charge there from far away, by any " \15             "path; the charge brought in times that potential is the work."16end

playground.rb

1require_relative "../solutions/bringing_in_a_charge"23play_with :bringing_in_a_charge do4  title "Bringing In a Charge"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 source	$V_{gs}$	V	—
charge of source	$q_{s}$	C	—
distance of ground from source	$r_{gs}$	m	—
potential of ground by brought	$V_{gb}$	V	—
charge of brought	$q_{b}$	C	—
distance of ground from brought	$r_{gb}$	m	—
potential of source by ground	$V_{sg}$	V	—
charge of ground	$q_{g}$	C	—
distance of source from ground	$r_{s}$	m	—
potential of source by brought	$V_{sb}$	V	—
distance of source from brought	$r_{sb}$	m	—
potential of brought by ground	$V_{bg}$	V	—
distance of brought from ground	$r_{b}$	m	—
potential of brought by source	$V_{bs}$	V	—
distance of brought from source	$r_{bs}$	m	—
work to bring in on ground by source	$W_{gs}$	J	—
work to bring in on ground by brought	$W_{gb}$	J	—
work to bring in on source by ground	$W_{sg}$	J	—
work to bring in on source by brought	$W_{sb}$	J	—
work to bring in on brought by ground	$W_{bg}$	J	—
work to bring in on brought by source	$W_{bs}$	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_{gs} = \frac{k \cdot q_{s}}{r_{gs}}

potential of a point charge(ground, source)

V_{gb} = \frac{k \cdot q_{b}}{r_{gb}}

potential of a point charge(ground, brought)

V_{sg} = \frac{k \cdot q_{g}}{r_{s}}

potential of a point charge(source, ground)

V_{sb} = \frac{k \cdot q_{b}}{r_{sb}}

potential of a point charge(source, brought)

V_{bg} = \frac{k \cdot q_{g}}{r_{b}}

potential of a point charge(brought, ground)

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

potential of a point charge(brought, source)

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

work is charge times potential(ground, source)

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

work is charge times potential(ground, brought)

W_{sg} = q_{s} \cdot V_{sg}

work is charge times potential(source, ground)

W_{sb} = q_{s} \cdot V_{sb}

work is charge times potential(source, brought)

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

work is charge times potential(brought, ground)

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

work is charge times potential(brought, source)