SME
Electromagnetic Theory · Problem type
A toroid: the field inside its windings
Use this for “A toroid of 800 turns carries 1.5 A. Find the field 12 cm from its center, inside the windings.”
Diagram
B
=
μ₀
NI
2πr
B
=
1.26 ×
10
-6
×
500
×
2
2π × 0.1
B = 0.002 T
The Amperian circle outside the ring encloses no net current, so B = 0 there.
This is an example. Type your own values to start a new problem.
B (
)
=
μ₀
×
I (
)
2π
r (
)
Formulas
B
=
4π
×
10
-7
×
N
×
I
2π ×
r
0.002
=
4π
×
10
-7
×
500
×
2
2π × 0.1
Formulas are worked in m, T. The step-by-step shows the conversions.
Assumptions
•
The turns are wound evenly and closely round a ring, so B runs round the ring in circles; the core is air, μ₀ = 4π × 10⁻⁷ T·m/A.
•
Ampère’s circle of radius r inside the windings threads every turn once: B × 2πr = μ₀NI.
•
In the hole and outside the ring the enclosed current adds to 0, so B = 0 there.
Step-by-step
WE KNOW
N = 500, I = 2 A, r = 10 cm
FIND
field at r (B)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m, T).
r = 10 cm = 0.1 m (1 m = 100 cm)
Step 2 · Find field at r (B)
Use B = 4π × 10⁻⁷ × N × I ÷ (2π × r)
Ampère’s
law
round
the
circle
of
radius
r
:
it
threads
N
wires,
each
carrying
I
.
B
=
4π
×
10
-7
×
500
×
2
2π × 0.1
B
=
0.002
T
Step 3 · Convert the answers
Change each answer back to the unit you chose.
B = 0.002 T = 2 mT (1 T = 1000 mT)
Check
Put the numbers back in. Both sides should match.
0.002
=
4π
×
10
-7
×
500
×
2
2π × 0.1
✓
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Ampère’s law: B inside and outside a thick wire
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A coil in a uniform field: its moment, torque and energy
Topic check
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Related lessons
Magnetostatics
Main lesson
Ampère’s law: B inside and outside a thick wire
Use this for “A wire of radius 3 mm carries 30 A spread evenly over its cross-section. Find B 2 mm and 6 mm from its axis.”
A coil in a uniform field: its moment, torque and energy
Use this for “A 120-turn coil of area 25 cm² carries 0.4 A in a 0.3 T field, its moment at 60° to the field. Find the torque and the energy.”