A Charge Above a Square

solved GausssLaw

Q. A point charge 10 μC $q_{c} = 10$ μC is a distance 5 cm directly above the centre of a square of side 10 cm. What is the magnitude of the electric flux through the square, in N m²/C? (Hint: Think of the square as one face of a cube $Φ_{s} = \frac{Φ_{c}}{6}$ with edge 10 cm.)

Inputs

charge

q_{c}

10 μC

Outputs

flux

Φ_{s}

How it's solved

Solved with GausssLaw

Given

$q_{c} = 10$ μC
$Φ_{s} = \frac{Φ_{c}}{6}$ from “one face of a cube”

Find $Φ_{s}$ .

Working

$q_{c} = 10$ μC $= 1 \times 10^{−5}$ C
gausss law(cube) $Φ_{c} = \frac{q_{c}}{ε_{0}}$
$Φ_{c} = \frac{1 \times 10^{−5}}{8.85418781 \times 10^{−12}}$
$Φ_{c} = 1.12940907 \times 10^{6}$ N m²/C
one face of a cube $Φ_{s} = \frac{Φ_{c}}{6}$
$Φ_{s} = \frac{1.12940907 \times 10^{6}}{6}$
$Φ_{s} = 188200$ N m²/C
$Φ_{s} \approx 1.88 \times 10^{5}$ N m²/Cto 3 significant figures

Conclusion

The charge sits at the centre of a cube with the square as one face, so the flux through the cube shares out equally among its six faces.

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_above_a_square do4  kind :numerical5  describes "A point charge %{charge} is a distance 5 cm directly above the centre of a square of side 10 cm. " \6            "%{flux}? (Hint: Think of the square as one face of a cube with edge 10 cm.)"7  given charge: "10 μC"8  asks flux: "What is the magnitude of the electric flux through the square, in N m²/C"9end

solution.rb

1require_relative "../questions/a_charge_above_a_square"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :a_charge_above_a_square, using: GausssLaw do5  objects :square, :cube67  take charge: cube.charge_inside89  deduce_from "one face of a cube", square.electric_flux => cube.electric_flux / 61011  flux = square.electric_flux12  answer flux:, in: "N m2/C", figures: 31314  conclusion "The charge sits at the centre of a cube with the square as one face, so the flux through the cube " \15             "shares out equally among its six faces."16end

playground.rb

1require_relative "../solutions/a_charge_above_a_square"23play_with :a_charge_above_a_square do4  title "A Charge Above a Square"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 square	$A_{s}$	m²	—
radius of square	$r_{s}$	m	0 … ∞
area of cube	$A_{c}$	m²	—
radius of cube	$r_{c}$	m	0 … ∞
electric flux of ground	$Φ_{g}$	N m²/C	—
electric flux of square	$Φ_{s}$	N m²/C	—
electric flux of cube	$Φ_{c}$	N m²/C	—
charge inside of ground	$q_{g}$	C	—
charge inside of square	$q_{s}$	C	—
charge inside of cube	$q_{c}$	C	—
diameter of ground	$D_{g}$	m	0 … ∞
diameter of square	$D_{s}$	m	0 … ∞
diameter of cube	$D_{c}$	m	0 … ∞
surface charge density of ground	$σ_{g}$	C/m²	—
surface charge density of square	$σ_{s}$	C/m²	—
surface charge density of cube	$σ_{c}$	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 square	$E_{l,s}$	V/m	—
linear charge density of square	$λ_{s}$	C/m	—
field near a line of cube	$E_{l,c}$	V/m	—
linear charge density of cube	$λ_{c}$	C/m	—
field between plates of ground	$E_{p,g}$	V/m	—
field between plates of square	$E_{p,s}$	V/m	—
field between plates of cube	$E_{p,c}$	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_{s} = π \cdot {r_{s}}^{2}

a round face(square)

A_{c} = π \cdot {r_{c}}^{2}

a round face(cube)

Φ_{g} = E \cdot A_{g}

flux through a face across the field(ground)

Φ_{s} = E \cdot A_{s}

flux through a face across the field(square)

Φ_{c} = E \cdot A_{c}

flux through a face across the field(cube)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ_{s} = \frac{q_{s}}{ε_{0}}

gausss law(square)

Φ_{c} = \frac{q_{c}}{ε_{0}}

gausss law(cube)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A_{s} = π \cdot {D_{s}}^{2}

the surface of a sphere(square)

A_{c} = π \cdot {D_{c}}^{2}

the surface of a sphere(cube)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q_{s} = σ_{s} \cdot A_{s}

charge spread over a surface(square)

q_{c} = σ_{c} \cdot A_{c}

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,s} = \frac{λ_{s}}{2 \cdot π \cdot ε_{0} \cdot r_{s}}

field of a long line of charge(square)

E_{l,c} = \frac{λ_{c}}{2 \cdot π \cdot ε_{0} \cdot r_{c}}

field of a long line of charge(cube)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p,s} = \frac{σ_{s}}{ε_{0}}

field between charged plates(square)

E_{p,c} = \frac{σ_{c}}{ε_{0}}

field between charged plates(cube)

Φ_{s} = \frac{Φ_{c}}{6}

one face of a cube