A Cylinder in a Field

solved GausssLaw

Q. An electric field is uniform, of magnitude 200 N/C $E = 200$ N/C, pointing away from the plane x = 0 on either side $Φ_{c} = 2 \cdot Φ_{f}$ . A right circular cylinder of radius 5 cm $r_{f} = 5$ cm has its centre at the origin and its axis along the x-axis, so that its flat faces stand across the field on either side. What is the net charge inside the cylinder, in coulombs?

Inputs

field

E

200 N/C

radius

r_{f}

5 cm

Outputs

charge

q_{c}

How it's solved

Solved with GausssLaw

Given

$E = 200$ N/C
$r_{f} = 5$ cm
$Φ_{c} = 2 \cdot Φ_{f}$ from “on either side”

Find $q_{c}$ .

Working

$E = 200$ N/C $= 200$ V/m
$r_{f} = 5$ cm $= 0.05$ m
a round face(face) $A_{f} = π \cdot {r_{f}}^{2}$
$A_{f} = 3.14159265 \cdot {0.05}^{2}$
$A_{f} = 0.00785398163$ m²
flux through a face across the field(face) $Φ_{f} = E \cdot A_{f}$
$Φ_{f} = 200 \cdot 0.00785398163$
$Φ_{f} = 1.57079633$ N m²/C
on either side $Φ_{c} = 2 \cdot Φ_{f}$
$Φ_{c} = 2 \cdot 1.57079633$
$Φ_{c} = 3.14159265$ N m²/C
gausss law(cylinder) $Φ_{c} = \frac{q_{c}}{ε_{0}}$
$3.14159265 = \frac{q_{c}}{8.85418781 \times 10^{−12}}$
$q_{c} = 2.782 \times 10^{−11}$ C
$q_{c} \approx 2.78 \times 10^{−11}$ Cto 3 significant figures

Conclusion

The field leaves through both flat faces and runs along the curved side, so all the flux is through the faces; Gauss's law turns that flux into the charge inside.

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_cylinder_in_a_field do4  kind :numerical5  describes "An electric field is uniform, of magnitude %{field}, pointing away from the plane x = 0 on either side. " \6            "A right circular cylinder of radius %{radius} has its centre at the origin and its axis along the " \7            "x-axis, so that its flat faces stand across the field on either side. %{charge}, in coulombs?"8  given field: "200 N/C"9  given radius: "5 cm"10  asks charge: "What is the net charge inside the cylinder"11end

solution.rb

1require_relative "../questions/a_cylinder_in_a_field"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :a_cylinder_in_a_field, using: GausssLaw do5  objects :face, :cylinder67  take field: E8  take radius: face.radius910  deduce_from "on either side", cylinder.electric_flux => 2 * face.electric_flux1112  charge = cylinder.charge_inside13  answer charge:, in: "C", figures: 31415  conclusion "The field leaves through both flat faces and runs along the curved side, so all the flux is " \16             "through the faces; Gauss's law turns that flux into the charge inside."17end

playground.rb

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

a round face(face)

A_{c} = π \cdot {r_{c}}^{2}

a round face(cylinder)

Φ_{g} = E \cdot A_{g}

flux through a face across the field(ground)

Φ_{f} = E \cdot A_{f}

flux through a face across the field(face)

Φ_{c} = E \cdot A_{c}

flux through a face across the field(cylinder)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ_{f} = \frac{q_{f}}{ε_{0}}

gausss law(face)

Φ_{c} = \frac{q_{c}}{ε_{0}}

gausss law(cylinder)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A_{f} = π \cdot {D_{f}}^{2}

the surface of a sphere(face)

A_{c} = π \cdot {D_{c}}^{2}

the surface of a sphere(cylinder)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q_{f} = σ_{f} \cdot A_{f}

charge spread over a surface(face)

q_{c} = σ_{c} \cdot A_{c}

charge spread over a surface(cylinder)

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

field of a long line of charge(face)

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

field of a long line of charge(cylinder)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p,f} = \frac{σ_{f}}{ε_{0}}

field between charged plates(face)

E_{p,c} = \frac{σ_{c}}{ε_{0}}

field between charged plates(cylinder)

Φ_{c} = 2 \cdot Φ_{f}

on either side