A Charge Above a Square

Q. A point charge 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 with edge 10 cm.) expects 1.88 × 10⁵ N m²/C

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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.)

The playground

Inputs

charge

q_{c}

10 μC

Outputs

flux

Φ_{s}

The solution

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 Ruby

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

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

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

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