Copper Spheres Doubled and Halved

solved CoulombsLaw

Q. Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm $r_{ab} = 50$ cm, and the charge on each $q_{b} = q_{a}$ is 6.5 × 10⁻⁷ C $q_{a} = 6.5 \times 10^{−7}$ C. The radii of A and B are negligible compared to the distance of separation. What is the force of repulsion if each sphere is charged double the above amount $q_{A2} = 2 \cdot q_{a}$ , and the distance between them is halved $r_{A2B2} = \frac{r_{ab}}{2}$ , in newtons?

Inputs

apart

r_{ab}

50 cm

charge

q_{a}

6.5e-07 C

Outputs

force

F_{e,A2B2}

How it's solved

Solved with CoulombsLaw

Given

$r_{ab} = 50$ cm
$q_{a} = 6.5 \times 10^{−7}$ C
$q_{b} = q_{a}$ from “the charge on each”
$q_{A2} = 2 \cdot q_{a}$ from “charged double the above amount”
$q_{B2} = 2 \cdot q_{b}$ from “each sphere is charged double”
$r_{A2B2} = \frac{r_{ab}}{2}$ from “the distance between them is halved”

Find $F_{e,A2B2}$ .

Working

$r_{ab} = 50$ cm $= 0.5$ m
charged double the above amount $q_{A2} = 2 \cdot q_{a}$
$q_{A2} = 2 \cdot 6.5 \times 10^{−7}$
$q_{A2} = 1.3 \times 10^{−6}$ C
the charge on each $q_{b} = q_{a}$
$q_{b} = 6.5 \times 10^{−7}$
$q_{b} = 6.5 \times 10^{−7}$ C
each sphere is charged double $q_{B2} = 2 \cdot q_{b}$
$q_{B2} = 2 \cdot 6.5 \times 10^{−7}$
$q_{B2} = 1.3 \times 10^{−6}$ C
the distance between them is halved $r_{A2B2} = \frac{r_{ab}}{2}$
$r_{A2B2} = \frac{0.5}{2}$
$r_{A2B2} = 0.25$ m
coulombs law(a after, b after) $F_{e,A2B2} = \frac{k \cdot q_{A2} \cdot q_{B2}}{{r_{A2B2}}^{2}}$
$F_{e,A2B2} = \frac{8.98755179 \times 10^{9} \cdot 1.3 \times 10^{−6} \cdot 1.3 \times 10^{−6}}{{0.25}^{2}}$
$F_{e,A2B2} = 0.243$ N
$F_{e,A2B2} \approx 0.24$ Nto 2 significant figures

Conclusion

Doubling each charge makes the force four times as strong, and halving the distance makes it four times stronger again.

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 :copper_spheres_doubled_and_halved do4  kind :numerical5  describes "Two insulated charged copper spheres A and B have their centres separated by a distance of %{apart}, " \6            "and the charge on each is %{charge}. The radii of A and B are negligible compared to the distance of " \7            "separation. %{force} if each sphere is charged double the above amount, and the distance between " \8            "them is halved, in newtons?"9  given apart: "50 cm"10  given charge: "6.5e-7 C"11  asks force: "What is the force of repulsion"12end

solution.rb

1require_relative "../questions/copper_spheres_doubled_and_halved"2require_relative "../science/electric_charges_and_fields/coulombs_law"34solve :copper_spheres_doubled_and_halved, using: CoulombsLaw do5  objects :a, :b, a_after: :A2, b_after: :B267  take apart: a.separation(from: b)8  take charge: a.charge910  deduce_from "the charge on each", b.charge => a.charge11  deduce_from "charged double the above amount", a_after.charge => 2 * a.charge12  deduce_from "each sphere is charged double", b_after.charge => 2 * b.charge13  deduce_from "the distance between them is halved", a_after.separation(from: b_after) => a.separation(from: b) / 21415  force = a_after.electric_force(by: b_after)16  answer force:, in: "N", figures: 21718  conclusion "Doubling each charge makes the force four times as strong, and halving the distance makes it four " \19             "times stronger again."20end

playground.rb

1require_relative "../solutions/copper_spheres_doubled_and_halved"23play_with :copper_spheres_doubled_and_halved do4  title "Copper Spheres Doubled and Halved"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 a	$F_{e,ga}$	N	—
charge of ground	$q_{g}$	C	—
charge of a	$q_{a}$	C	—
separation of ground from a	$r_{ga}$	m	0 … ∞
electric force on ground by b	$F_{e,gb}$	N	—
charge of b	$q_{b}$	C	—
separation of ground from b	$r_{gb}$	m	0 … ∞
electric force on ground by a after	$F_{e,gA2}$	N	—
charge of a after	$q_{A2}$	C	—
separation of ground from a after	$r_{gA2}$	m	0 … ∞
electric force on ground by b after	$F_{e,gB2}$	N	—
charge of b after	$q_{B2}$	C	—
separation of ground from b after	$r_{gB2}$	m	0 … ∞
electric force on a by ground	$F_{e,ag}$	N	—
separation of a from ground	$r_{a}$	m	0 … ∞
electric force on a by b	$F_{e,ab}$	N	—
separation of a from b	$r_{ab}$	m	0 … ∞
electric force on a by a after	$F_{e,aA2}$	N	—
separation of a from a after	$r_{aA2}$	m	0 … ∞
electric force on a by b after	$F_{e,aB2}$	N	—
separation of a from b after	$r_{aB2}$	m	0 … ∞
electric force on b by ground	$F_{e,bg}$	N	—
separation of b from ground	$r_{b}$	m	0 … ∞
electric force on b by a	$F_{e,ba}$	N	—
separation of b from a	$r_{ba}$	m	0 … ∞
electric force on b by a after	$F_{e,bA2}$	N	—
separation of b from a after	$r_{bA2}$	m	0 … ∞
electric force on b by b after	$F_{e,bB2}$	N	—
separation of b from b after	$r_{bB2}$	m	0 … ∞
electric force on a after by ground	$F_{e,A2g}$	N	—
separation of a after from ground	$r_{A2}$	m	0 … ∞
electric force on a after by a	$F_{e,A2a}$	N	—
separation of a after from a	$r_{A2a}$	m	0 … ∞
electric force on a after by b	$F_{e,A2b}$	N	—
separation of a after from b	$r_{A2b}$	m	0 … ∞
electric force on a after by b after	$F_{e,A2B2}$	N	—
separation of a after from b after	$r_{A2B2}$	m	0 … ∞
electric force on b after by ground	$F_{e,B2g}$	N	—
separation of b after from ground	$r_{B2}$	m	0 … ∞
electric force on b after by a	$F_{e,B2a}$	N	—
separation of b after from a	$r_{B2a}$	m	0 … ∞
electric force on b after by b	$F_{e,B2b}$	N	—
separation of b after from b	$r_{B2b}$	m	0 … ∞
electric force on b after by a after	$F_{e,B2A2}$	N	—
separation of b after from a after	$r_{B2A2}$	m	0 … ∞

F_{e,oo} = \frac{k \cdot q_{o} \cdot q_{o}}{{r_{oo}}^{2}}

coulombs law

F_{e,ga} = \frac{k \cdot q_{g} \cdot q_{a}}{{r_{ga}}^{2}}

coulombs law(ground, a)

F_{e,gb} = \frac{k \cdot q_{g} \cdot q_{b}}{{r_{gb}}^{2}}

coulombs law(ground, b)

F_{e,gA2} = \frac{k \cdot q_{g} \cdot q_{A2}}{{r_{gA2}}^{2}}

coulombs law(ground, a after)

F_{e,gB2} = \frac{k \cdot q_{g} \cdot q_{B2}}{{r_{gB2}}^{2}}

coulombs law(ground, b after)

F_{e,ag} = \frac{k \cdot q_{a} \cdot q_{g}}{{r_{a}}^{2}}

coulombs law(a, ground)

F_{e,ab} = \frac{k \cdot q_{a} \cdot q_{b}}{{r_{ab}}^{2}}

coulombs law(a, b)

F_{e,aA2} = \frac{k \cdot q_{a} \cdot q_{A2}}{{r_{aA2}}^{2}}

coulombs law(a, a after)

F_{e,aB2} = \frac{k \cdot q_{a} \cdot q_{B2}}{{r_{aB2}}^{2}}

coulombs law(a, b after)

F_{e,bg} = \frac{k \cdot q_{b} \cdot q_{g}}{{r_{b}}^{2}}

coulombs law(b, ground)

F_{e,ba} = \frac{k \cdot q_{b} \cdot q_{a}}{{r_{ba}}^{2}}

coulombs law(b, a)

F_{e,bA2} = \frac{k \cdot q_{b} \cdot q_{A2}}{{r_{bA2}}^{2}}

coulombs law(b, a after)

F_{e,bB2} = \frac{k \cdot q_{b} \cdot q_{B2}}{{r_{bB2}}^{2}}

coulombs law(b, b after)

F_{e,A2g} = \frac{k \cdot q_{A2} \cdot q_{g}}{{r_{A2}}^{2}}

coulombs law(a after, ground)

F_{e,A2a} = \frac{k \cdot q_{A2} \cdot q_{a}}{{r_{A2a}}^{2}}

coulombs law(a after, a)

F_{e,A2b} = \frac{k \cdot q_{A2} \cdot q_{b}}{{r_{A2b}}^{2}}

coulombs law(a after, b)

F_{e,A2B2} = \frac{k \cdot q_{A2} \cdot q_{B2}}{{r_{A2B2}}^{2}}

coulombs law(a after, b after)

F_{e,B2g} = \frac{k \cdot q_{B2} \cdot q_{g}}{{r_{B2}}^{2}}

coulombs law(b after, ground)

F_{e,B2a} = \frac{k \cdot q_{B2} \cdot q_{a}}{{r_{B2a}}^{2}}

coulombs law(b after, a)

F_{e,B2b} = \frac{k \cdot q_{B2} \cdot q_{b}}{{r_{B2b}}^{2}}

coulombs law(b after, b)

F_{e,B2A2} = \frac{k \cdot q_{B2} \cdot q_{A2}}{{r_{B2A2}}^{2}}

coulombs law(b after, a after)

q_{b} = q_{a}

the charge on each

q_{A2} = 2 \cdot q_{a}

charged double the above amount

q_{B2} = 2 \cdot q_{b}

each sphere is charged double

r_{A2B2} = \frac{r_{ab}}{2}

the distance between them is halved