SME
Engineering Electromagnetics · Problem type
Skin depth in a conductor
Use this for “Find the skin depth of copper (σ = 5.8 × 10⁷ S/m) at 1 MHz.”
Diagram
δ
=
1
÷
√
πfμσ
δ
=
1
÷
√
π × 1 ×
10
6
× 1.257 ×
10
-6
× 5.8 ×
10
7
δ
=
6.609 ×
10
-5
m
Jz
J₀
=
e
−z/δ
=
e
−1.513
Jz
J₀
=
22.02%
The current crowds into the skin: the shading fades by e every δ in from the surface.
This is an example. Type your own values to start a new problem.
δ (μm)
=
1
π
f (MHz)
×
μ₀
σ (S/m)
Jz
%
=
100%
×
e
−
μm
/
μm
Formulas
δ
=
1
÷
√
π ×
f
× 4π ×
10
-7
× μᵣ ×
σ
6.6085 ×
10
-5
=
1
÷
√(π
×
1,000,000
×
4π
×
10
-7
×
1
×
(5.8
×
10
7
))
Jz
=
100
×
e
−
z
÷
δ
22.0206
=
100
×
e
−0.0001 ÷ (6.6085 ×
10
-5
)
Formulas are worked in Hz, m. The step-by-step shows the conversions.
Assumptions
•
A good conductor (σ far above ωε) with a flat surface much thicker than δ.
•
μ = μrμ₀ with μ₀ = 4π × 10⁻⁷ H/m; copper and aluminium have μᵣ = 1.
•
The current density falls as Jz = J₀e−z/δ below the surface; Jz is given as a percent of J₀.
Step-by-step
WE KNOW
f = 1 MHz, σ = 5.8 × 10⁷ S/m, μᵣ = 1, z = 100 μm
FIND
skin depth (δ), current density left (Jz)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to Hz, m).
f = 1 MHz = 1,000,000 Hz (1 MHz = 10⁶ Hz)
z = 100 μm = 0.0001 m (1 m = 10⁶ μm)
Step 2 · Find skin depth (δ)
Use δ = 1 ÷ √(π × f × 4π × 10⁻⁷ × μᵣ × σ)
The
field
loses
a
factor
e
every
δ
;
faster
changes
and
better
conductors
shrink
δ
.
δ
=
1
÷
√(π
×
1,000,000
×
4π
×
10
-7
×
1
×
(5.8
×
10
7
))
δ
=
6.6085 ×
10
-5
m
Step 3 · Find current density left (Jz)
Use Jz = 100 × e−z ÷ δ
Each
skin
depth
down,
the
current
density
drops
to
1
e
,
about
36.8%,
of
what
it
was.
Jz
=
100
×
e
−0.0001 ÷ (6.60855 ×
10
-5
)
Jz
=
100
×
e
−1.513
Jz = 100 × 0.2202
Jz = 22.0206%
Step 4 · Convert the answers
Change each answer back to the unit you chose.
δ = 6.6085 × 10⁻⁵ m = 66.08549 μm (1 m = 10⁶ μm)
Check
Put the numbers back in. Both sides should match.
6.60855 ×
10
-5
=
1
÷
√(π
×
1,000,000
×
4π
×
10
-7
×
1
×
(5.8
×
10
7
))
✓
22.0206
=
100
×
e
−0.0001 ÷ (6.60855 ×
10
-5
)
✓
Topic check
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