Two Spheres Touched and Brought Closer

solved CoulombsLaw

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?

Outputs

ratio?

How it's solved

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 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 :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

solution.rb

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

playground.rb

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 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_{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