SME
University Physics II: Electricity & Magnetism · Problem type
Two parallel wires: the field and the force between them
Use this for “Two long parallel wires 8 cm apart carry 12 A and 6 A in opposite directions. Find the field of the first at the second and the force per meter.”
Diagram
B
=
μ₀
I
2πr
=
1.26 ×
10
-6
×
10
2π × 0.05
B
=
4 ×
10
-5
T
F/L
=
μ₀I₁
I₂
2πr
=
4 ×
10
-4
N/m,
pulling
the
wires
together
This is an example. Type your own values to start a new problem.
B (
)
=
μ₀
I₁ (
)
2π
r (
)
I₂
Current in wire 2
example
F/L
Force per length on each wire
calculated · N/m
Formulas
B
=
4π
×
10
-7
×
I₁
2π ×
r
4 ×
10
-5
=
4π
×
10
-7
×
10
2π × 0.05
F/L
=
I₂
×
B
+4
×
10
-4
=
10
×
(4
×
10
-5
)
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
The wires are long, straight and parallel, r apart center to center, in air with μ₀ = 4π × 10⁻⁷ T·m/A.
•
Ampère’s law round a circle of radius r about wire 1 gives B × 2πr = μ₀I₁, so B = μ₀I₁ ÷ (2πr), circling the wire.
•
Wire 2 sits in that field, so each meter of it feels F/L = I₂B; by Newton’s third law wire 1 feels the same force back.
•
A positive I₂ runs the same way as I₁ and the wires attract (F/L > 0); a negative I₂ runs the other way and they repel.
Step-by-step
WE KNOW
I₁ = 10 A, r = 5 cm, I₂ = +10 A
FIND
field of wire 1 at wire 2 (B), force per length on each wire (F/L)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m).
r = 5 cm = 0.05 m (1 m = 100 cm)
Step 2 · Find field of wire 1 at wire 2 (B)
Use B = 4π × 10⁻⁷ × I₁ ÷ (2π × r)
Ampère’s
law
round
a
circle
of
radius
r
about
wire
1:
B
×
2πr
=
μ₀
I₁
.
B
=
4π
×
10
-7
×
10
2π × 0.05
B
=
4 ×
10
-5
T
Step 3 · Find force per length on each wire (F/L)
Use F/L = I₂ × B
The
force
on
each
meter
of
wire
2
in
wire
1’s
field,
μ₀
I₁
I₂
2πr
;
+
pulls
the
wires
together.
F/L
=
10
×
(4
×
10
-5
)
F/L
=
+4
×
10
-4
N/m
Check
Put the numbers back in. Both sides should match.
4 ×
10
-5
=
4π
×
10
-7
×
10
2π × 0.05
✓
+4
×
10
-4
=
10
×
(4
×
10
-5
)
✓
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