The Flux Leaving a Charged Sphere

solved GausssLaw

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. What is the total electric flux leaving the surface of the sphere, in N m²/C?

Inputs

diameter

D

2.40 m

density

σ

80 μC/m²

Outputs

flux

Φ

How it's solved

Solved with GausssLaw

Given

$D = 2.4$ m
$σ = 80.0$ μC/m²

Find $Φ$ .

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) $q = σ \cdot A$
$q = 8 \times 10^{−5} \cdot 18.0955737$
$q = 0.00144764589$ C
gausss law(sphere) $Φ = \frac{q}{ε_{0}}$
$Φ = \frac{0.00144764589}{8.85418781 \times 10^{−12}}$
$Φ = 1.635 \times 10^{8}$ N m²/C
$Φ \approx 1.6 \times 10^{8}$ N m²/Cto 2 significant figures

Conclusion

The charge on the sphere is its density times its area, and by Gauss's law all of it sends its flux out through the surface.

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 :the_flux_leaving_a_charged_sphere do4  kind :numerical5  describes "A uniformly charged conducting sphere of %{diameter} diameter has a surface charge density of " \6            "%{density}. %{flux}, in N m²/C?"7  given diameter: "2.4 m"8  given density: "80.0 μC/m2"9  asks flux: "What is the total electric flux leaving the surface of the sphere"10end

solution.rb

1require_relative "../questions/the_flux_leaving_a_charged_sphere"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :the_flux_leaving_a_charged_sphere, using: GausssLaw do5  objects :sphere67  take diameter: sphere.diameter8  take density: sphere.surface_charge_density910  flux = sphere.electric_flux11  answer flux:, in: "N m2/C", figures: 21213  conclusion "The charge on the sphere is its density times its area, and by Gauss's law all of it sends its " \14             "flux out through the surface."15end

playground.rb

1require_relative "../solutions/the_flux_leaving_a_charged_sphere"23play_with :the_flux_leaving_a_charged_sphere do4  title "The Flux Leaving a Charged Sphere"5end

The laws it drew on

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)