Two Parallel Wires

Q. Two long and parallel straight wires A and B carrying currents of 8.0 A and 5.0 A in the same direction are separated by a distance of 4.0 cm. Estimate the force on a 10 cm section of wire A, in newtons? expects 2 × 10⁻⁵ N

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The web of concepts

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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?

The playground

Inputs

current a

I_{a}

8 A

current b

I_{b}

5 A

apart

d_{b}

4 cm

Outputs

force

F_{a}

The solution

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 Ruby

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

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

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

The laws

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