Two Spheres Touched and Brought Closer

Q. A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B is brought close to A such that the distance between their centres is 10 cm, and the repulsion of A is noted. Spheres A and B are touched by identical uncharged spheres C and D respectively, which are then removed, and B is brought closer to A to a distance of 5.0 cm between their centres. How many times the first repulsion is the repulsion now? expects 1.0

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Q. A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B is brought close to A such that the distance between their centres is 10 cm, and the repulsion of A is noted. Spheres A and B are touched by identical uncharged spheres $q_{A2} = \frac{q_{a}}{2}$ , $q_{B2} = \frac{q_{b}}{2}$ C and D respectively, which are then removed, and B is brought closer to A to a distance of 5.0 cm $r_{A2B2} = \frac{r_{ab}}{2}$ between their centres. How many times the first repulsion is the repulsion now?

The playground

Outputs

ratio?

The solution

Solved with CoulombsLaw

Given

$q_{A2} = \frac{q_{a}}{2}$ from “touched by identical uncharged spheres”
$q_{B2} = \frac{q_{b}}{2}$ from “touched by identical uncharged spheres”
$r_{A2B2} = \frac{r_{ab}}{2}$ from “brought closer to A to a distance of 5.0 cm”

Find $\frac{F_{e,A2B2}}{F_{e,ab}}$ .

Working

coulombs law(a, b) $F_{e,ab} = \frac{k \cdot q_{a} \cdot q_{b}}{{r_{ab}}^{2}}$
$F_{e,ab} = \frac{8.98755179 \times 10^{9} \cdot q_{a} \cdot q_{b}}{{r_{ab}}^{2}}$
coulombs law(a after, b after) $F_{e,A2B2} = \frac{k \cdot q_{A2} \cdot q_{B2}}{{r_{A2B2}}^{2}}$
$F_{e,A2B2} = \frac{8.98755179 \times 10^{9} \cdot \frac{q_{a}}{2} \cdot \frac{q_{b}}{2}}{{(\frac{r_{ab}}{2})}^{2}}$
$\frac{F_{e,A2B2}}{F_{e,ab}} = 1$

Conclusion

Touching an identical uncharged sphere shares the charge equally, halving each; halving the distance quadruples the force again, so the two changes cancel.

The Ruby

1require_relative "../physics"23question :two_spheres_touched_and_brought_closer do4  kind :numerical5  describes "A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B is brought " \6            "close to A such that the distance between their centres is 10 cm, and the repulsion of A is noted. " \7            "Spheres A and B are touched by identical uncharged spheres C and D respectively, which are then " \8            "removed, and B is brought closer to A to a distance of 5.0 cm between their centres. %{ratio}?"9  asks ratio: "How many times the first repulsion is the repulsion now"10end

1require_relative "../questions/two_spheres_touched_and_brought_closer"2require_relative "../science/electric_charges_and_fields/coulombs_law"34solve :two_spheres_touched_and_brought_closer, using: CoulombsLaw do5  objects :a, :b, a_after: :A2, b_after: :B267  deduce_from "touched by identical uncharged spheres", a_after.charge => a.charge / 2, b_after.charge => b.charge / 28  deduce_from "brought closer to A to a distance of 5.0 cm", a_after.separation(from: b_after) => a.separation(from: b) / 2910  answer find_relation(a_after.electric_force(by: b_after) / a.electric_force(by: b)), figures: 21112  conclusion "Touching an identical uncharged sphere shares the charge equally, halving each; halving the " \13             "distance quadruples the force again, so the two changes cancel."14end

1require_relative "../solutions/two_spheres_touched_and_brought_closer"23play_with :two_spheres_touched_and_brought_closer do4  title "Two Spheres Touched and Brought Closer"5end

The laws

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_{A2} = \frac{q_{a}}{2}

touched by identical uncharged spheres (charge of a after)

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

touched by identical uncharged spheres (charge of b after)

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

brought closer to A to a distance of 5.0 cm