Between Two Charged Plates

solved GausssLaw

Q. Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10⁻²² C/m² $σ = 17.0 \times 10^{−22}$ C/m². What is E between the plates, in N/C?

Inputs

density

σ

1.7e-21 C/m²

Outputs

field

E_{p}

How it's solved

Solved with GausssLaw

Given

$σ = 17.0 \times 10^{−22}$ C/m²

Find $E_{p}$ .

Working

field between charged plates(plates) $E_{p} = \frac{σ}{ε_{0}}$
$E_{p} = \frac{1.7 \times 10^{−21}}{8.85418781 \times 10^{−12}}$
$E_{p} = 1.92 \times 10^{−10}$ N/C
$E_{p} = 1.91999541 \times 10^{−10}$ V/m $= 1.91999541 \times 10^{−10}$ N/C
$E_{p} \approx 1.92 \times 10^{−10}$ N/Cto 3 significant figures

Conclusion

Between the plates the fields of the two opposite sheets of charge add, giving the charge density over the permittivity of free space; outside they cancel.

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 :between_two_charged_plates do4  kind :numerical5  describes "Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates " \6            "have surface charge densities of opposite signs and of magnitude %{density}. %{field}, in N/C?"7  given density: "17.0e-22 C/m2"8  asks field: "What is E between the plates"9end

solution.rb

1require_relative "../questions/between_two_charged_plates"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :between_two_charged_plates, using: GausssLaw do5  objects :plates67  take density: plates.surface_charge_density89  field = plates.field_between_plates10  answer field:, in: "N/C", figures: 31112  conclusion "Between the plates the fields of the two opposite sheets of charge add, giving the charge density " \13             "over the permittivity of free space; outside they cancel."14end

playground.rb

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

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(plates)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ = \frac{q}{ε_{0}}

gausss law(plates)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A = π \cdot D^{2}

the surface of a sphere(plates)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q = σ \cdot A

charge spread over a surface(plates)

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(plates)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p} = \frac{σ}{ε_{0}}

field between charged plates(plates)