Bringing In a Charge

Q. A charge of 4 × 10⁻⁷ C is located 9 cm from a point P. What is the work done in bringing a charge of 2 × 10⁻⁹ C from infinity to the point P, in joules? expects 8.0 × 10⁻⁵ J

What the draft establishes

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The web of concepts

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The equations it used

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Check an answer

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?

The playground

Inputs

source

q_{s}

4e-07 C

distance

r_{bs}

9 cm

brought

q_{b}

2e-09 C

Outputs

work

W_{bs}

The solution

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 Ruby

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

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

1require_relative "../solutions/bringing_in_a_charge"23play_with :bringing_in_a_charge do4  title "Bringing In a Charge"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 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)