A Charge in a Gaussian Cube

solved GausssLaw

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?

Inputs

charge

q

2 μC

Outputs

flux

Φ

How it's solved

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 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 :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

solution.rb

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

playground.rb

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 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 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)