A Cylinder in a Field

Q. An electric field is uniform, of magnitude 200 N/C, pointing away from the plane x = 0 on either side. A right circular cylinder of radius 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? expects 2.78 × 10⁻¹¹ C

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The web of concepts

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The equations it used

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Check an answer

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?

The playground

Inputs

field

E

200 N/C

radius

r_{f}

5 cm

Outputs

charge

q_{c}

The solution

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 Ruby

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

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

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

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