Flux out of a Black Box

solved GausssLaw

Q. Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 10³ N m²/C $Φ = 8.0 \times 10^{3}$ N m²/C. What is the net charge inside the box, in μC?

Inputs

flux

Φ

8000 N m²/C

Outputs

charge

q

How it's solved

Solved with GausssLaw

Given

$Φ = 8.0 \times 10^{3}$ N m²/C

Find $q$ .

Working

gausss law(box) $Φ = \frac{q}{ε_{0}}$
$8000 = \frac{q}{8.85418781 \times 10^{−12}}$
$q = 0.07083$ μC
$q = 7.08335025 \times 10^{−8}$ C $= 0.0708335025$ μC
$q \approx 0.07$ μCto 1 significant figure

Conclusion

Gauss's law ties the net flux out of any closed surface to the charge inside it, whatever the shape of the box.

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 :flux_out_of_a_black_box do4  kind :numerical5  describes "Careful measurement of the electric field at the surface of a black box indicates that the net outward " \6            "flux through the surface of the box is %{flux}. %{charge}, in μC?"7  given flux: "8.0e3 N m2/C"8  asks charge: "What is the net charge inside the box"9end

solution.rb

1require_relative "../questions/flux_out_of_a_black_box"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :flux_out_of_a_black_box, using: GausssLaw do5  objects :box67  take flux: box.electric_flux89  charge = box.charge_inside10  answer charge:, in: "μC", figures: 11112  conclusion "Gauss's law ties the net flux out of any closed surface to the charge inside it, whatever the " \13             "shape of the box."14end

playground.rb

1require_relative "../solutions/flux_out_of_a_black_box"23play_with :flux_out_of_a_black_box do4  title "Flux out of a Black Box"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 box	$A$	m²	—
radius of box	$r$	m	0 … ∞
electric flux of ground	$Φ_{g}$	N m²/C	—
electric flux of box	$Φ$	N m²/C	—
charge inside of ground	$q_{g}$	C	—
charge inside of box	$q$	C	—
diameter of ground	$D_{g}$	m	0 … ∞
diameter of box	$D$	m	0 … ∞
surface charge density of ground	$σ_{g}$	C/m²	—
surface charge density of box	$σ$	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 box	$E_{l}$	V/m	—
linear charge density of box	$λ$	C/m	—
field between plates of ground	$E_{p,g}$	V/m	—
field between plates of box	$E_{p}$	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(box)

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(box)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ = \frac{q}{ε_{0}}

gausss law(box)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A = π \cdot D^{2}

the surface of a sphere(box)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q = σ \cdot A

charge spread over a surface(box)

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(box)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p} = \frac{σ}{ε_{0}}

field between charged plates(box)