A Solenoid with an Iron Core

Q. A solenoid has a core of a material with relative permeability 400. The windings of the solenoid are insulated from the core and carry a current of 2 A. If the number of turns is 1000 m⁻¹, what is the magnetic field in the core, in teslas? expects 1.0 T

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Q. A solenoid has a core of a material with relative permeability 400 $μ_{r} = 400$ . The windings of the solenoid are insulated from the core and carry a current of 2 A $I = 2$ A. If the number of turns is 1000 m⁻¹ $n = 1000$ m⁻¹, what is the magnetic field in the core, in teslas?

The playground

Inputs

permeability

μ_{r}

400

current

I

2 A

turns

n

1000 m⁻¹

Outputs

field

B

The solution

Solved with Magnetisation

Given

$μ_{r} = 400$
$I = 2$ A
$n = 1000$ m⁻¹

Find $B$ .

Working

intensity of a solenoid(core) $H = n \cdot I$
$H = 1000 \cdot 2$
$H = 2000$ A/m
field in a magnetised core(core) $B = μ_{r} \cdot μ_{0} \cdot H$
$B = 400 \cdot 1.25663706 \times 10^{−6} \cdot 2000$
$B = 1.005$ T
$B \approx 1.0$ Tto 2 significant figures

Conclusion

The windings alone set H; the core lines up its own atomic magnets along it, multiplying the field four hundred times over.

The Ruby

1require_relative "../physics"23question :a_solenoid_with_an_iron_core do4  kind :numerical5  describes "A solenoid has a core of a material with relative permeability %{permeability}. The windings of the " \6            "solenoid are insulated from the core and carry a current of %{current}. If the number of turns is " \7            "%{turns}, %{field}, in teslas?"8  given permeability: "400"9  given current: "2 A"10  given turns: "1000 m-1"11  asks field: "what is the magnetic field in the core"12end

1require_relative "../questions/a_solenoid_with_an_iron_core"2require_relative "../science/magnetism_and_matter/magnetisation"34solve :a_solenoid_with_an_iron_core, using: Magnetisation do5  objects :core67  take permeability: core.relative_permeability8  take current: core.current9  take turns: core.turns_per_length1011  field = core.magnetic_field12  answer field:, in: "T", figures: 21314  conclusion "The windings alone set H; the core lines up its own atomic magnets along it, multiplying the " \15             "field four hundred times over."16end

1require_relative "../solutions/a_solenoid_with_an_iron_core"23play_with :a_solenoid_with_an_iron_core do4  title "A Solenoid with an Iron Core"5end

The laws

quantity	written	units	within	from
permeability of free space	$μ_{0}$	N/A²	—	Magnetisation
relative permeability	$μ_{r}$	—	—	Magnetisation
current	$I$	A	—	Magnetisation
turns per length	$n$	m⁻¹	—	Magnetisation
magnetic intensity	$H$	A/m	—	Magnetisation
magnetisation	$M$	A/m	—	Magnetisation
magnetic field	$B$	T	—	Magnetisation
magnetising current	$I_{m}$	A	—	Magnetisation
magnetic intensity of ground	$H_{g}$	A/m	—
turns per length of ground	$n_{g}$	m⁻¹	—
current of ground	$I_{g}$	A	—
magnetic intensity of core	$H$	A/m	—
turns per length of core	$n$	m⁻¹	—
current of core	$I$	A	—
magnetisation of ground	$M_{g}$	A/m	—
relative permeability of ground	$μ_{r,g}$	—	—
magnetisation of core	$M$	A/m	—
relative permeability of core	$μ_{r}$	—	—
magnetic field of ground	$B_{g}$	T	—
magnetic field of core	$B$	T	—
magnetising current of ground	$I_{m,g}$	A	—
magnetising current of core	$I_{m}$	A	—

H_{c} = n_{c} \cdot I_{c}

intensity of a solenoid

M_{c} = (μ_{r,c} - 1) \cdot H_{c}

magnetisation of a core

B_{c} = μ_{r,c} \cdot μ_{0} \cdot H_{c}

field in a magnetised core

B_{c} = μ_{0} \cdot n_{c} \cdot (I_{c} + I_{m,c})

the current that would do without the core

H_{g} = n_{g} \cdot I_{g}

intensity of a solenoid(ground)

H = n \cdot I

intensity of a solenoid(core)

M_{g} = (μ_{r,g} - 1) \cdot H_{g}

magnetisation of a core(ground)

M = (μ_{r} - 1) \cdot H

magnetisation of a core(core)

B_{g} = μ_{r,g} \cdot μ_{0} \cdot H_{g}

field in a magnetised core(ground)

B = μ_{r} \cdot μ_{0} \cdot H

field in a magnetised core(core)

B_{g} = μ_{0} \cdot n_{g} \cdot (I_{g} + I_{m,g})

the current that would do without the core(ground)

B = μ_{0} \cdot n \cdot (I + I_{m})

the current that would do without the core(core)