Two Parallel Wires

solved MagneticFieldOfCurrents · MagneticForce

Q. Two long and parallel straight wires $B = B_{w,b}$ A and B carrying currents of 8.0 A $I_{a} = 8.0$ A and 5.0 A $I_{b} = 5.0$ A in the same direction are separated by a distance of 4.0 cm $d_{b} = 4.0$ cm. Estimate the force on a 10 cm section of wire A $l_{a} = 10$ cm, in newtons?

Inputs

current a

I_{a}

8 A

current b

I_{b}

5 A

apart

d_{b}

4 cm

Outputs

force

F_{a}

How it's solved

Solved with MagneticFieldOfCurrents, MagneticForce

Given

$I_{a} = 8.0$ A
$I_{b} = 5.0$ A
$d_{b} = 4.0$ cm
$l_{a} = 10$ cm from “a 10 cm section of wire A”
$B = B_{w,b}$ from “parallel straight wires”

Find $F_{a}$ .

Working

$d_{b} = 4.0$ cm $= 0.04$ m
$l_{a} = 10$ cm $= 0.1$ m
field near a long straight wire(b) $B_{w,b} = \frac{μ_{0} \cdot I_{b}}{2 \cdot π \cdot d_{b}}$
$B_{w,b} = \frac{1.25663706 \times 10^{−6} \cdot 5}{2 \cdot 3.14159265 \cdot 0.04}$
$B_{w,b} = 2.5 \times 10^{−5}$ T
parallel straight wires ${magnetic}_{field} = B_{w,b}$
${magnetic}_{field} = 2.5 \times 10^{−5}$
$B = 2.5 \times 10^{−5}$ T
force on a wire across a field(a) $F_{a} = I_{a} \cdot l_{a} \cdot B$
$F_{a} = 8 \cdot 0.1 \cdot 2.5 \times 10^{−5}$
$F_{a} = 2 \times 10^{−5}$ N
$F_{a} \approx 2 \times 10^{−5}$ Nto 1 significant figure

Conclusion

Wire B's field circles it and crosses wire A square on, pushing on A's current; currents in the same direction pull together.

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 :two_parallel_wires do4  kind :numerical5  describes "Two long and parallel straight wires A and B carrying currents of %{current_a} and %{current_b} in the " \6            "same direction are separated by a distance of %{apart}. %{force}, in newtons?"7  given current_a: "8.0 A"8  given current_b: "5.0 A"9  given apart: "4.0 cm"10  asks force: "Estimate the force on a 10 cm section of wire A"11end

solution.rb

1require_relative "../questions/two_parallel_wires"2require_relative "../science/moving_charges_and_magnetism/magnetic_field_of_currents"3require_relative "../science/moving_charges_and_magnetism/magnetic_force"45solve :two_parallel_wires, using: [ MagneticFieldOfCurrents, MagneticForce ] do6  objects :a, :b78  take current_a: a.current9  take current_b: b.current10  take apart: b.distance_from_the_wire1112  deduce_from "parallel straight wires", B => b.field_near_a_wire13  deduce_from "a 10 cm section of wire A", a.length_in_the_field => "10 cm"1415  force = a.force_on_a_wire16  answer force:, in: "N", figures: 11718  conclusion "Wire B's field circles it and crosses wire A square on, pushing on A's current; currents in the " \19             "same direction pull together."20end

playground.rb

1require_relative "../solutions/two_parallel_wires"23play_with :two_parallel_wires do4  title "Two Parallel Wires"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	MagneticFieldOfCurrents
permeability of free space	$μ_{0}$	N/A²	—	MagneticFieldOfCurrents
current	$I$	A	—	MagneticFieldOfCurrents
turns	$N$	—	—	MagneticFieldOfCurrents
radius	$R$	m	—	MagneticFieldOfCurrents
length of the coil	$L$	m	—	MagneticFieldOfCurrents
turns per length	$n$	m⁻¹	—	MagneticFieldOfCurrents
field at the centre	$B_{c}$	T	—	MagneticFieldOfCurrents
field inside	$B_{s}$	T	—	MagneticFieldOfCurrents
distance from the wire	$d$	m	—	MagneticFieldOfCurrents
field near a wire	$B_{w}$	T	—	MagneticFieldOfCurrents
elementary charge	$e$	C	—	MagneticForce
gravity	$g$	m/s²	—	MagneticForce
magnetic field	$B$	T	—	MagneticForce
mass	$m$	kg	0 … ∞	MagneticForce
charge	$q$	C	—	MagneticForce
speed	$v$	m/s	—	MagneticForce
length in the field	$l$	m	—	MagneticForce
force on a wire	$F$	N	—	MagneticForce
force per length	$f$	N/m	—	MagneticForce
radius of the path	$r$	m	—	MagneticForce
angle to the field	$θ$	—	0 … 3.142	MagneticForce
frequency of revolution	$ν$	Hz	—	MagneticForce
field at the centre of ground	$B_{c,g}$	T	—
turns of ground	$N_{g}$	—	—
current of ground	$I_{g}$	A	—
radius of ground	$R_{g}$	m	—
field at the centre of a	$B_{c,a}$	T	—
turns of a	$N_{a}$	—	—
current of a	$I_{a}$	A	—
radius of a	$R_{a}$	m	—
field at the centre of b	$B_{c,b}$	T	—
turns of b	$N_{b}$	—	—
current of b	$I_{b}$	A	—
radius of b	$R_{b}$	m	—
turns per length of ground	$n_{g}$	m⁻¹	—
length of the coil of ground	$L_{g}$	m	—
turns per length of a	$n_{a}$	m⁻¹	—
length of the coil of a	$L_{a}$	m	—
turns per length of b	$n_{b}$	m⁻¹	—
length of the coil of b	$L_{b}$	m	—
field near a wire of ground	$B_{w,g}$	T	—
distance from the wire of ground	$d_{g}$	m	—
field near a wire of a	$B_{w,a}$	T	—
distance from the wire of a	$d_{a}$	m	—
field near a wire of b	$B_{w,b}$	T	—
distance from the wire of b	$d_{b}$	m	—
field inside of ground	$B_{s,g}$	T	—
field inside of a	$B_{s,a}$	T	—
field inside of b	$B_{s,b}$	T	—
force on a wire on ground	$F_{g}$	N	—
length in the field of ground	$l_{g}$	m	—
force on a wire on a	$F_{a}$	N	—
length in the field of a	$l_{a}$	m	—
force on a wire on b	$F_{b}$	N	—
length in the field of b	$l_{b}$	m	—
force per length on ground	$f_{g}$	N/m	—
angle to the field of ground	$θ_{g}$	—	0 … 3.142
force per length on a	$f_{a}$	N/m	—
angle to the field of a	$θ_{a}$	—	0 … 3.142
force per length on b	$f_{b}$	N/m	—
angle to the field of b	$θ_{b}$	—	0 … 3.142
radius of the path of ground	$r_{g}$	m	—
mass of ground	$m_{g}$	kg	0 … ∞
speed of ground	$v_{g}$	m/s	—
charge of ground	$q_{g}$	C	—
radius of the path of a	$r_{a}$	m	—
mass of a	$m_{a}$	kg	0 … ∞
speed of a	$v_{a}$	m/s	—
charge of a	$q_{a}$	C	—
radius of the path of b	$r_{b}$	m	—
mass of b	$m_{b}$	kg	0 … ∞
speed of b	$v_{b}$	m/s	—
charge of b	$q_{b}$	C	—
frequency of revolution of ground	$ν_{g}$	Hz	—
frequency of revolution of a	$ν_{a}$	Hz	—
frequency of revolution of b	$ν_{b}$	Hz	—

B_{c,c} = \frac{μ_{0} \cdot N_{c} \cdot I_{c}}{2 \cdot R_{c}}

field at the centre of a coil

n_{c} = \frac{N_{c}}{L_{c}}

turns on each metre

B_{w,w} = \frac{μ_{0} \cdot I_{w}}{2 \cdot π \cdot d_{w}}

field near a long straight wire

B_{s,c} = μ_{0} \cdot n_{c} \cdot I_{c}

field inside a long solenoid

F_{w} = I_{w} \cdot l_{w} \cdot B

force on a wire across a field

f_{w} = I_{w} \cdot B \cdot \sin θ_{w}

force on each metre of wire

r_{c} = \frac{m_{c} \cdot v_{c}}{q_{c} \cdot B}

a charge circles in a field

v_{c} = 2 \cdot π \cdot r_{c} \cdot ν_{c}

once round the circle

B_{c,g} = \frac{μ_{0} \cdot N_{g} \cdot I_{g}}{2 \cdot R_{g}}

field at the centre of a coil(ground)

B_{c,a} = \frac{μ_{0} \cdot N_{a} \cdot I_{a}}{2 \cdot R_{a}}

field at the centre of a coil(a)

B_{c,b} = \frac{μ_{0} \cdot N_{b} \cdot I_{b}}{2 \cdot R_{b}}

field at the centre of a coil(b)

n_{g} = \frac{N_{g}}{L_{g}}

turns on each metre(ground)

n_{a} = \frac{N_{a}}{L_{a}}

turns on each metre(a)

n_{b} = \frac{N_{b}}{L_{b}}

turns on each metre(b)

B_{w,g} = \frac{μ_{0} \cdot I_{g}}{2 \cdot π \cdot d_{g}}

field near a long straight wire(ground)

B_{w,a} = \frac{μ_{0} \cdot I_{a}}{2 \cdot π \cdot d_{a}}

field near a long straight wire(a)

B_{w,b} = \frac{μ_{0} \cdot I_{b}}{2 \cdot π \cdot d_{b}}

field near a long straight wire(b)

B_{s,g} = μ_{0} \cdot n_{g} \cdot I_{g}

field inside a long solenoid(ground)

B_{s,a} = μ_{0} \cdot n_{a} \cdot I_{a}

field inside a long solenoid(a)

B_{s,b} = μ_{0} \cdot n_{b} \cdot I_{b}

field inside a long solenoid(b)

F_{g} = I_{g} \cdot l_{g} \cdot B

force on a wire across a field(ground)

F_{a} = I_{a} \cdot l_{a} \cdot B

force on a wire across a field(a)

F_{b} = I_{b} \cdot l_{b} \cdot B

force on a wire across a field(b)

f_{g} = I_{g} \cdot B \cdot \sin θ_{g}

force on each metre of wire(ground)

f_{a} = I_{a} \cdot B \cdot \sin θ_{a}

force on each metre of wire(a)

f_{b} = I_{b} \cdot B \cdot \sin θ_{b}

force on each metre of wire(b)

r_{g} = \frac{m_{g} \cdot v_{g}}{q_{g} \cdot B}

a charge circles in a field(ground)

r_{a} = \frac{m_{a} \cdot v_{a}}{q_{a} \cdot B}

a charge circles in a field(a)

r_{b} = \frac{m_{b} \cdot v_{b}}{q_{b} \cdot B}

a charge circles in a field(b)

v_{g} = 2 \cdot π \cdot r_{g} \cdot ν_{g}

once round the circle(ground)

v_{a} = 2 \cdot π \cdot r_{a} \cdot ν_{a}

once round the circle(a)

v_{b} = 2 \cdot π \cdot r_{b} \cdot ν_{b}

once round the circle(b)

{magnetic}_{field} = B_{w,b}

parallel straight wires