A Sphere with an Inward Field

solved ElectricField

Q. A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm $r_{ps} = 20$ cm from the centre of the sphere is 1.5 × 10³ N/C $E = 1.5 \times 10^{3}$ N/C and points radially inward $E_{q,ps} = - E$ , what is the net charge on the sphere, in nC?

Inputs

distance

r_{ps}

20 cm

field

E

1500 N/C

Outputs

charge

q_{s}

How it's solved

Solved with ElectricField

Given

$r_{ps} = 20$ cm
$E = 1.5 \times 10^{3}$ N/C
$E_{q,ps} = - E$ from “points radially inward”

Find $q_{s}$ .

Working

$r_{ps} = 20$ cm $= 0.2$ m
$E = 1.5 \times 10^{3}$ N/C $= 1500$ V/m
points radially inward $E_{q,ps} = - E$
$E_{q,ps} = - 1500$
$E_{q,ps} = −1500$ V/m
field of a point charge(point, sphere) $E_{q,ps} = \frac{k \cdot q_{s}}{{r_{ps}}^{2}}$
$(−1500) = \frac{8.98755179 \times 10^{9} \cdot q_{s}}{{0.2}^{2}}$
$q_{s} = −6.676$ nC
$q_{s} = −6.67590033 \times 10^{−9}$ C $= −6.67590033$ nC
$q_{s} \approx −6.68$ nCto 3 significant figures

Conclusion

Outside a charged sphere the field is as if all the charge sat at its centre; a field pointing inward means the charge is negative.

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 :a_sphere_with_an_inward_field do4  kind :numerical5  describes "A conducting sphere of radius 10 cm has an unknown charge. If the electric field %{distance} from the " \6            "centre of the sphere is %{field} and points radially inward, %{charge}, in nC?"7  given distance: "20 cm"8  given field: "1.5e3 N/C"9  asks charge: "what is the net charge on the sphere"10end

solution.rb

1require_relative "../questions/a_sphere_with_an_inward_field"2require_relative "../science/electric_charges_and_fields/electric_field"34solve :a_sphere_with_an_inward_field, using: ElectricField do5  objects :sphere, :point67  take distance: point.separation(from: sphere)8  take field: E910  deduce_from "points radially inward", point.field_of_a_charge(by: sphere) => -E1112  charge = sphere.charge13  answer charge:, in: "nC", figures: 31415  conclusion "Outside a charged sphere the field is as if all the charge sat at its centre; a field pointing " \16             "inward means the charge is negative."17end

playground.rb

1require_relative "../solutions/a_sphere_with_an_inward_field"23play_with :a_sphere_with_an_inward_field do4  title "A Sphere with an Inward Field"5end

The laws it drew on

quantity	written	units	within	from
charge	$q$	C	—	ElectricField
coulomb constant	$k$	N m² C⁻²	—	ElectricField
field strength	$E$	V/m	—	ElectricField
separation	$r$	m	0 … ∞	ElectricField
field of a charge	$E_{q}$	V/m	—	ElectricField
force of the field	$F$	N	—	ElectricField
force of the field on ground	$F_{g}$	N	—
charge of ground	$q_{g}$	C	—
force of the field on sphere	$F_{s}$	N	—
charge of sphere	$q_{s}$	C	—
force of the field on point	$F_{p}$	N	—
charge of point	$q_{p}$	C	—
field of a charge of ground by sphere	$E_{q,gs}$	V/m	—
separation of ground from sphere	$r_{gs}$	m	0 … ∞
field of a charge of ground by point	$E_{q,gp}$	V/m	—
separation of ground from point	$r_{gp}$	m	0 … ∞
field of a charge of sphere by ground	$E_{q,sg}$	V/m	—
separation of sphere from ground	$r_{s}$	m	0 … ∞
field of a charge of sphere by point	$E_{q,sp}$	V/m	—
separation of sphere from point	$r_{sp}$	m	0 … ∞
field of a charge of point by ground	$E_{q,pg}$	V/m	—
separation of point from ground	$r_{p}$	m	0 … ∞
field of a charge of point by sphere	$E_{q,ps}$	V/m	—
separation of point from sphere	$r_{ps}$	m	0 … ∞

F_{b} = q_{b} \cdot E

force on a charge in a field

E_{q,ps} = \frac{k \cdot q_{s}}{{r_{ps}}^{2}}

field of a point charge

F_{g} = q_{g} \cdot E

force on a charge in a field(ground)

F_{s} = q_{s} \cdot E

force on a charge in a field(sphere)

F_{p} = q_{p} \cdot E

force on a charge in a field(point)

E_{q,gs} = \frac{k \cdot q_{s}}{{r_{gs}}^{2}}

field of a point charge(ground, sphere)

E_{q,gp} = \frac{k \cdot q_{p}}{{r_{gp}}^{2}}

field of a point charge(ground, point)

E_{q,sg} = \frac{k \cdot q_{g}}{{r_{s}}^{2}}

field of a point charge(sphere, ground)

E_{q,sp} = \frac{k \cdot q_{p}}{{r_{sp}}^{2}}

field of a point charge(sphere, point)

E_{q,pg} = \frac{k \cdot q_{g}}{{r_{p}}^{2}}

field of a point charge(point, ground)

E_{q,ps} = \frac{k \cdot q_{s}}{{r_{ps}}^{2}}

field of a point charge(point, sphere)

E_{q,ps} = - E

points radially inward