An Infinite Line Charge

Q. An infinite line charge produces a field of 9 × 10⁴ N/C at a distance of 2 cm. Calculate the linear charge density, in C/m? expects 1.0 × 10⁻⁷ C/m

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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?

The playground

Inputs

field

E_{l}

9e+04 N/C

distance

r

2 cm

Outputs

density

λ

The solution

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 Ruby

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

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

1require_relative "../solutions/an_infinite_line_charge"23play_with :an_infinite_line_charge do4  title "An Infinite Line Charge"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 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)