An Iron Core's Magnetising Current

solved Magnetisation

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 magnetising current, the extra current that would give the same field without the core, in amperes?

Inputs

permeability

μ_{r}

400

current

I

2 A

turns

n

1000 m⁻¹

Outputs

current needed

I_{m}

How it's solved

Solved with Magnetisation

Given

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

Find $I_{m}$ .

Working

intensity of a solenoid(core) $H = n \cdot I$
$H = 1000 \cdot 2$
$H = 2000$ A/m
field in a magnetised core(core) $B = μ_{r} \cdot μ_{0} \cdot H$
$B = 400 \cdot 1.25663706 \times 10^{−6} \cdot 2000$
$B = 1.00530965$ T
the current that would do without the core(core) $B = μ_{0} \cdot n \cdot (I + I_{m})$
$1.00530965 = 1.25663706 \times 10^{−6} \cdot 1000 \cdot (2 + I_{m})$
$I_{m} = 798$ A
$I_{m} \approx 798$ Ato 3 significant figures

Conclusion

Without the core the windings would need hundreds of amperes more to make the same field; the core's atoms supply it.

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_iron_core_s_magnetising_current 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}, %{current_needed}, in amperes?"8  given permeability: "400"9  given current: "2 A"10  given turns: "1000 m-1"11  asks current_needed: "what is the magnetising current, the extra current that would give the same field without the core"12end

solution.rb

1require_relative "../questions/an_iron_core_s_magnetising_current"2require_relative "../science/magnetism_and_matter/magnetisation"34solve :an_iron_core_s_magnetising_current, using: Magnetisation do5  objects :core67  take permeability: core.relative_permeability8  take current: core.current9  take turns: core.turns_per_length1011  current_needed = core.magnetising_current12  answer current_needed:, in: "A", figures: 31314  conclusion "Without the core the windings would need hundreds of amperes more to make the same field; the core's atoms supply it."15end

playground.rb

1require_relative "../solutions/an_iron_core_s_magnetising_current"23play_with :an_iron_core_s_magnetising_current do4  title "An Iron Core's Magnetising Current"5end

The laws it drew on

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)