A Coil of a Hundred Small Turns

Q. A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the centre of the coil, in teslas? expects 3.1 × 10⁻⁴ T

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Q. A circular coil of wire consisting of 100 $N = 100$ turns, each of radius 8.0 cm $R = 8.0$ cm carries a current of 0.40 A $I = 0.40$ A. What is the magnitude of the magnetic field B at the centre of the coil, in teslas?

The playground

Inputs

turns

N

100

radius

R

8 cm

current

I

0.4 A

Outputs

field

B_{c}

The solution

Solved with MagneticFieldOfCurrents

Given

$N = 100$
$R = 8.0$ cm
$I = 0.40$ A

Find $B_{c}$ .

Working

$R = 8.0$ cm $= 0.08$ m
field at the centre of a coil(coil) $B_{c} = \frac{μ_{0} \cdot N \cdot I}{2 \cdot R}$
$B_{c} = \frac{1.25663706 \times 10^{−6} \cdot 100 \cdot 0.4}{2 \cdot 0.08}$
$B_{c} = 3.142 \times 10^{−4}$ T
$B_{c} \approx 3.1 \times 10^{−4}$ Tto 2 significant figures

Conclusion

Every turn adds its own field at the centre, so the field grows with the turns and the current and falls as the coil gets wider.

The Ruby

1require_relative "../physics"23question :a_coil_of_a_hundred_small_turns do4  kind :numerical5  describes "A circular coil of wire consisting of %{turns} turns, each of radius %{radius} carries a current of " \6            "%{current}. %{field}, in teslas?"7  given turns: "100"8  given radius: "8.0 cm"9  given current: "0.40 A"10  asks field: "What is the magnitude of the magnetic field B at the centre of the coil"11end

1require_relative "../questions/a_coil_of_a_hundred_small_turns"2require_relative "../science/moving_charges_and_magnetism/magnetic_field_of_currents"34solve :a_coil_of_a_hundred_small_turns, using: MagneticFieldOfCurrents do5  objects :coil67  take turns: coil.turns8  take radius: coil.radius9  take current: coil.current1011  field = coil.field_at_the_centre12  answer field:, in: "T", figures: 21314  conclusion "Every turn adds its own field at the centre, so the field grows with the turns and the current " \15             "and falls as the coil gets wider."16end

1require_relative "../solutions/a_coil_of_a_hundred_small_turns"23play_with :a_coil_of_a_hundred_small_turns do4  title "A Coil of a Hundred Small Turns"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
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 coil	$B_{c}$	T	—
turns of coil	$N$	—	—
current of coil	$I$	A	—
radius of coil	$R$	m	—
turns per length of ground	$n_{g}$	m⁻¹	—
length of the coil of ground	$L_{g}$	m	—
turns per length of coil	$n$	m⁻¹	—
length of the coil of coil	$L$	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 coil	$B_{w}$	T	—
distance from the wire of coil	$d$	m	—
field inside of ground	$B_{s,g}$	T	—
field inside of coil	$B_{s}$	T	—

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

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} = \frac{μ_{0} \cdot N \cdot I}{2 \cdot R}

field at the centre of a coil(coil)

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

turns on each metre(ground)

n = \frac{N}{L}

turns on each metre(coil)

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

field near a long straight wire(ground)

B_{w} = \frac{μ_{0} \cdot I}{2 \cdot π \cdot d}

field near a long straight wire(coil)

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

field inside a long solenoid(ground)

B_{s} = μ_{0} \cdot n \cdot I

field inside a long solenoid(coil)