SME
University Physics II: Electricity & Magnetism
Magnetic fields and induction
Use this for “A 50-turn coil of area 20 cm² has 4 Ω. The field through it falls from 0.8 T to 0.2 T in 0.05 s. Find the emf and the current.”
Diagram
ε = NΔΦ/Δt = 200 × 0.004/0.2 = 4 V
Pushing the N pole in, the coil’s current makes its near end an N pole that pushes back (Lenz’s law).
Faster motion or more turns: a bigger emf.
No motion: no current.
Area of each turn A = 0.01 m²
Field at the start B₁ = +0.1 T
Field at the end B₂ = +0.5 T
Resistance of the coil R = 8 Ω
Induced current I = 0.5 A
This is an example. Type your own values to start a new problem.
ΔΦ (
)
=
|
B₂ (
)
−
B₁ (
)
|
×
A (
)
ε (
)
=
N
×
ΔΦ
Δt (
)
R
Resistance of the coil
example
I
Induced current
calculated
Formulas
ΔΦ
=
|
B₂
−
B₁
|
×
A
0.004 = |0.5 − 0.1| × 0.01
ε
=
N
×
ΔΦ
Δt
4
=
200
×
0.004
0.2
I
=
ε
R
0.5
=
4
8
Assumptions
•
The field is square to the coil’s face and the area stays the same, so the flux through each turn is Φ = BA.
•
The field changes steadily, so Faraday’s law ε = −N dΦ/dt gives a constant emf of size NΔΦ ÷ Δt.
•
The sign is Lenz’s law, said in words: the current’s own field opposes the change. ε and I are given as sizes.
•
A negative field points the other way through the coil, so a field that reverses changes the flux by the sum of the two sizes.
Step-by-step
WE KNOW
N = 200, A = 0.01 m², B₁ = +0.1 T, B₂ = +0.5 T, Δt = 0.2 s, R = 8 Ω
FIND
change in flux through each turn (ΔΦ), induced emf (ε), induced current (I)
Step 1 · Find change in flux through each turn (ΔΦ)
Use ΔΦ = |B₂ − B₁| × A
The flux through one turn is BA, so it changes by the change in the field times the area.
ΔΦ = |0.5 − 0.1| × 0.01
ΔΦ = |0.4| × 0.01
ΔΦ = 0.4 × 0.01
ΔΦ = 0.004 Wb
Step 2 · Find induced emf (ε)
Use ε = N × ΔΦ ÷ Δt
Faraday’s
law:
each
turn
gets
an
emf
of
ΔΦ
Δt
,
and
the
N
turns
in
series
add
up.
ε
=
200
×
0.004
0.2
ε
=
0.8
0.2
ε
=
4
V
Step 3 · Find induced current (I)
Use I = ε ÷ R
Ohm’s law round the coil: the emf drives the current through the coil’s resistance.
I
=
4
8
I
=
0.5
A
Check
Put the numbers back in. Both sides should match.
0.004 = |0.5 − 0.1| × 0.01 ✓
4
=
200
×
0.004
0.2
✓
0.5
=
4
8
✓
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A battery’s internal resistance
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A rod sliding on rails in a magnetic field
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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