A Sphere with an Inward Field

Q. A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 10³ N/C and points radially inward, what is the net charge on the sphere, in nC? expects -6.68 nC

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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?

The playground

Inputs

distance

r_{ps}

20 cm

field

E

1500 N/C

Outputs

charge

q_{s}

The solution

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 Ruby

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

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

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

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