A Charge in a Gaussian Cube

Q. A point charge of 2.0 μC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface, in N m²/C? expects 2.26 × 10⁵ N m²/C

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Q. A point charge of 2.0 μC $q = 2.0$ μC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface, in N m²/C?

The playground

Inputs

charge

q

2 μC

Outputs

flux

Φ

The solution

Solved with GausssLaw

Given

$q = 2.0$ μC

Find $Φ$ .

Working

$q = 2.0$ μC $= 2 \times 10^{−6}$ C
gausss law(cube) $Φ = \frac{q}{ε_{0}}$
$Φ = \frac{2 \times 10^{−6}}{8.85418781 \times 10^{−12}}$
$Φ = 225900$ N m²/C
$Φ \approx 2.26 \times 10^{5}$ N m²/Cto 3 significant figures

Conclusion

The flux through a closed surface depends only on the charge inside, so the size of the cube does not matter.

The Ruby

1require_relative "../physics"23question :a_charge_in_a_gaussian_cube do4  kind :numerical5  describes "A point charge of %{charge} is at the centre of a cubic Gaussian surface 9.0 cm on edge. %{flux}, in " \6            "N m²/C?"7  given charge: "2.0 μC"8  asks flux: "What is the net electric flux through the surface"9end

1require_relative "../questions/a_charge_in_a_gaussian_cube"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :a_charge_in_a_gaussian_cube, using: GausssLaw do5  objects :cube67  take charge: cube.charge_inside89  flux = cube.electric_flux10  answer flux:, in: "N m2/C", figures: 31112  conclusion "The flux through a closed surface depends only on the charge inside, so the size of the cube does " \13             "not matter."14end

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

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(cube)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ = \frac{q}{ε_{0}}

gausss law(cube)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A = π \cdot D^{2}

the surface of a sphere(cube)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q = σ \cdot A

charge spread over a surface(cube)

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(cube)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p} = \frac{σ}{ε_{0}}

field between charged plates(cube)