Flux out of a Black Box

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. What is the net charge inside the box, in μC? expects 0.07 μC

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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?

The playground

Inputs

flux

Φ

8000 N m²/C

Outputs

charge

q

The solution

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 Ruby

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

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

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

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)