The Charge on a Charged Sphere

Q. A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 μC/m². Find the charge on the sphere, in coulombs? expects 0.00145 C

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Q. A uniformly charged conducting sphere of 2.4 m $D = 2.4$ m diameter has a surface charge density of 80.0 μC/m² $σ = 80.0$ μC/m². Find the charge on the sphere, in coulombs?

The playground

Inputs

diameter

D

2.40 m

density

σ

80 μC/m²

Outputs

charge

q

The solution

Solved with GausssLaw

Given

$D = 2.4$ m
$σ = 80.0$ μC/m²

Find $q$ .

Working

$σ = 80.0$ μC/m² $= 8 \times 10^{−5}$ C/m²
the surface of a sphere(sphere) $A = π \cdot D^{2}$
$A = 3.14159265 \cdot {2.4}^{2}$
$A = 18.0955737$ m²
charge spread over a surface(sphere) $q = σ \cdot A$
$q = 8 \times 10^{−5} \cdot 18.0955737$
$q = 0.001448$ C
$q \approx 0.00145$ Cto 3 significant figures

Conclusion

The charge is spread evenly over the surface, so it is the charge on each square metre times the area of the sphere.

The Ruby

1require_relative "../physics"23question :the_charge_on_a_charged_sphere do4  kind :numerical5  describes "A uniformly charged conducting sphere of %{diameter} diameter has a surface charge density of " \6            "%{density}. %{charge}, in coulombs?"7  given diameter: "2.4 m"8  given density: "80.0 μC/m2"9  asks charge: "Find the charge on the sphere"10end

1require_relative "../questions/the_charge_on_a_charged_sphere"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :the_charge_on_a_charged_sphere, using: GausssLaw do5  objects :sphere67  take diameter: sphere.diameter8  take density: sphere.surface_charge_density910  charge = sphere.charge_inside11  answer charge:, in: "C", figures: 31213  conclusion "The charge is spread evenly over the surface, so it is the charge on each square metre times the " \14             "area of the sphere."15end

1require_relative "../solutions/the_charge_on_a_charged_sphere"23play_with :the_charge_on_a_charged_sphere do4  title "The Charge on a Charged Sphere"5end

The laws

quantity	written	units	within	from
pi	$π$	—	0 … ∞	GausssLaw
permittivity of free space	$ε_{0}$	F/m	—	GausssLaw
field strength	$E$	V/m	—	GausssLaw
area	$A$	m²	—	GausssLaw
radius	$r$	m	0 … ∞	GausssLaw
electric flux	$Φ$	N m²/C	—	GausssLaw
charge inside	$q$	C	—	GausssLaw
diameter	$D$	m	0 … ∞	GausssLaw
surface charge density	$σ$	C/m²	—	GausssLaw
linear charge density	$λ$	C/m	—	GausssLaw
field near a line	$E_{l}$	V/m	—	GausssLaw
field between plates	$E_{p}$	V/m	—	GausssLaw
area of ground	$A_{g}$	m²	—
radius of ground	$r_{g}$	m	0 … ∞
area of sphere	$A$	m²	—
radius of sphere	$r$	m	0 … ∞
electric flux of ground	$Φ_{g}$	N m²/C	—
electric flux of sphere	$Φ$	N m²/C	—
charge inside of ground	$q_{g}$	C	—
charge inside of sphere	$q$	C	—
diameter of ground	$D_{g}$	m	0 … ∞
diameter of sphere	$D$	m	0 … ∞
surface charge density of ground	$σ_{g}$	C/m²	—
surface charge density of sphere	$σ$	C/m²	—
field near a line of ground	$E_{l,g}$	V/m	—
linear charge density of ground	$λ_{g}$	C/m	—
field near a line of sphere	$E_{l}$	V/m	—
linear charge density of sphere	$λ$	C/m	—
field between plates of ground	$E_{p,g}$	V/m	—
field between plates of sphere	$E_{p}$	V/m	—

A_{f} = π \cdot {r_{f}}^{2}

a round face

Φ_{f} = E \cdot A_{f}

flux through a face across the field

Φ_{s} = \frac{q_{s}}{ε_{0}}

gausss law

A_{s} = π \cdot {D_{s}}^{2}

the surface of a sphere

q_{s} = σ_{s} \cdot A_{s}

charge spread over a surface

E_{l,l} = \frac{λ_{l}}{2 \cdot π \cdot ε_{0} \cdot r_{l}}

field of a long line of charge

E_{p,p} = \frac{σ_{p}}{ε_{0}}

field between charged plates

A_{g} = π \cdot {r_{g}}^{2}

a round face(ground)

A = π \cdot r^{2}

a round face(sphere)

Φ_{g} = E \cdot A_{g}

flux through a face across the field(ground)

Φ = E \cdot A

flux through a face across the field(sphere)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ = \frac{q}{ε_{0}}

gausss law(sphere)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A = π \cdot D^{2}

the surface of a sphere(sphere)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q = σ \cdot A

charge spread over a surface(sphere)

E_{l,g} = \frac{λ_{g}}{2 \cdot π \cdot ε_{0} \cdot r_{g}}

field of a long line of charge(ground)

E_{l} = \frac{λ}{2 \cdot π \cdot ε_{0} \cdot r}

field of a long line of charge(sphere)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p} = \frac{σ}{ε_{0}}

field between charged plates(sphere)