An Infinite Line Charge

solved GausssLaw

Q. An infinite line charge produces a field of 9 × 10⁴ N/C $E_{l} = 9 \times 10^{4}$ N/C at a distance of 2 cm $r = 2$ cm. Calculate the linear charge density, in C/m?

Inputs

field

E_{l}

9e+04 N/C

distance

r

2 cm

Outputs

density

λ

How it's solved

Solved with GausssLaw

Given

$E_{l} = 9 \times 10^{4}$ N/C
$r = 2$ cm

Find $λ$ .

Working

$E_{l} = 9 \times 10^{4}$ N/C $= 90000$ V/m
$r = 2$ cm $= 0.02$ m
field of a long line of charge(line) $E_{l} = \frac{λ}{2 \cdot π \cdot ε_{0} \cdot r}$
$90000 = \frac{λ}{2 \cdot 3.14159265 \cdot 8.85418781 \times 10^{−12} \cdot 0.02}$
$λ = 1.001 \times 10^{−7}$ C/m
$λ \approx 1.0 \times 10^{−7}$ C/mto 2 significant figures

Conclusion

Gauss's law on a cylinder round the line shows the field falls off as one over the distance, so the field at one distance fixes the charge on each metre.

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 :an_infinite_line_charge do4  kind :numerical5  describes "An infinite line charge produces a field of %{field} at a distance of %{distance}. %{density}, in C/m?"6  given field: "9e4 N/C"7  given distance: "2 cm"8  asks density: "Calculate the linear charge density"9end

solution.rb

1require_relative "../questions/an_infinite_line_charge"2require_relative "../science/electric_charges_and_fields/gausss_law"34solve :an_infinite_line_charge, using: GausssLaw do5  objects :line67  take field: line.field_near_a_line8  take distance: line.radius910  density = line.linear_charge_density11  answer density:, in: "C/m", figures: 21213  conclusion "Gauss's law on a cylinder round the line shows the field falls off as one over the distance, so " \14             "the field at one distance fixes the charge on each metre."15end

playground.rb

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

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(line)

Φ_{g} = \frac{q_{g}}{ε_{0}}

gausss law(ground)

Φ = \frac{q}{ε_{0}}

gausss law(line)

A_{g} = π \cdot {D_{g}}^{2}

the surface of a sphere(ground)

A = π \cdot D^{2}

the surface of a sphere(line)

q_{g} = σ_{g} \cdot A_{g}

charge spread over a surface(ground)

q = σ \cdot A

charge spread over a surface(line)

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(line)

E_{p,g} = \frac{σ_{g}}{ε_{0}}

field between charged plates(ground)

E_{p} = \frac{σ}{ε_{0}}

field between charged plates(line)