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Lecture 13 : Displacement current
• Displacement current
• Maxwell’s complete equations
Recap (1)
• Maxwell’s Equations for the 𝐸-field and 𝐵-field generated by
stationary (i.e., not varying with time) charge density 𝜌 and
current density 𝐽 are:
𝝆
𝜵. 𝑬 =
𝜺𝟎
𝜵×𝑬=𝟎
𝜵. 𝑩 = 𝟎
𝜵 × 𝑩 = 𝝁𝟎 𝑱
Recap (2)
• These equations are modified in
the presence of time-varying
electromagnetic fields
• Faraday’s Law describes how a
changing magnetic flux Φ
through a circuit induces a
voltage in the circuit, 𝑉 = −
𝜕Φ
𝜕𝑡
• This phenomenon is known as
electromagnetic induction, and
leads to the equation
modification 𝜵 × 𝑬
𝝏𝑩
+
𝝏𝒕
=𝟎
Displacement current
• Maxwell’s Equations must also be modified in the presence of
time-varying electric fields. The missing ingredient can be
illustrated in a couple of ways.
• First, consider taking the divergence of Ampere’s Law 𝛻 × 𝐵 =
𝜇0 𝐽. Since 𝛻. 𝛻 × 𝐵 = 0, we find 𝛻. 𝐽 = 0
• However, charge conservation implies that 𝛻. 𝐽 +
𝜕𝜌
𝜕𝑡
=0
• This contradiction implies that Ampere’s law must be wrong for
time-varying currents!
Displacement current
• This is fixed by the modification 𝜵 × 𝑩 =
𝝏𝑬
𝝁𝟎 𝑱 + 𝝁𝟎 𝜺𝟎
𝝏𝒕
• Now, take the divergence of both sides. We obtain 𝜇0 𝛻. 𝐽 +
𝜕𝐸
𝜇0 𝜀0 𝛻. = 0. Cancelling 𝜇0 and
𝜕𝑡
𝜕
𝜕
, this becomes 𝛻. 𝐽 + 𝜀0
𝛻. 𝐸
𝜕𝑡
𝜕𝑡
• Now substituting in 𝛻. 𝐸 =
𝜌
,
𝜀0
swapping the order of 𝛻 and
=0
we find 𝛻. 𝐽
𝜕𝜌
+
𝜕𝑡
=0
• The new version of the equation is consistent with charge
conservation!
Displacement current
• For time-varying magnetic fields : 𝜵 × 𝑬 +
• For time-varying electric fields : 𝜵 × 𝑩 =
𝝏𝑩
𝝏𝒕
=𝟎
𝝏𝑬
𝝁𝟎 𝑱 + 𝝁𝟎 𝜺𝟎
𝝏𝒕
• 𝐸- and 𝐵-fields are symmetric : a changing 𝑩-field generates
an 𝑬-field, and a changing 𝑬-field generates a 𝑩-field
𝜕𝐸
𝜀0
𝜕𝑡
• The term
in the 2nd equation is known as the
displacement current, since it acts like a current density and is
equal to
𝜕𝐷
,
𝜕𝑡
in terms of the electric displacement 𝐷 = 𝜀0 𝐸
Displacement current
• Second, consider a capacitor discharging into a circuit:
• Consider applying Ampere’s Law 𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 to both
the plane surface and the bulging surface in the diagram
Displacement current
• The current enclosed by these two surfaces is different!
• The situation can be reconciled by
including the displacement current
enclosed by the bulging surface
• The electric field 𝐸 = 𝜎/𝜀0 in terms
of the charge density 𝜎
• If 𝐴 is the plate area, then 𝑄 = 𝜎𝐴
𝑑𝑄
𝑑𝐸
hence 𝐼 =
= 𝜀0 𝐴
𝑑𝑡
𝑑𝑡
• This is equal to the displacement
current enclosed by the surface!
Maxwell’s complete equations
• Maxwell’s Equations are now complete!
• In a vacuum:
𝝆
𝜵. 𝑬 =
𝜺𝟎
𝝏𝑩
𝜵×𝑬+
=𝟎
𝝏𝒕
𝜵. 𝑩 = 𝟎
𝝏𝑬
𝜵 × 𝑩 − 𝝁𝟎 𝜺 𝟎
= 𝝁𝟎 𝑱
𝝏𝒕
Maxwell’s complete equations
• More generally, in materials with relative permittivity 𝜀𝑟 and
relative permeability 𝜇𝑟 we can use the electric displacement
𝐷 = 𝜀𝑟 𝜀0 𝐸 and magnetic intensity 𝐻 = 𝐵/𝜇𝑟 𝜇0 :
𝜵. 𝑫 = 𝝆
𝝏𝑩
𝜵×𝑬+
=𝟎
𝝏𝒕
𝜵. 𝑩 = 𝟎
𝝏𝑫
𝜵×𝑯−
=𝑱
𝝏𝒕
Maxwell’s complete equations
• The equations can be expressed in either differential
form or integral form, and these are equivalent
Differential form
Integral form
Name or effect
𝛻. 𝐷 = 𝜌
𝐷. 𝑑 𝐴 = 𝑄𝑒𝑛𝑐
Gauss’s Law
𝜕𝐵
𝛻×𝐸+
=0
𝜕𝑡
𝛻. 𝐵 = 0
𝜕𝐷
𝛻×𝐻−
=𝐽
𝜕𝑡
𝜕
𝐸. 𝑑 𝑙 = −
𝜕𝑡
𝐵. 𝑑 𝐴
𝐵. 𝑑𝐴 = 0
𝜕
𝐻. 𝑑 𝑙 = 𝐼𝑒𝑛𝑐 +
𝜕𝑡
Electromagnetic
induction
No monopoles
𝐷. 𝑑 𝐴
Ampere’s Law,
Displacement current
Summary
• An electric field is generated by a changing magnetic
field (electromagnetic induction)
• A magnetic field is also generated by a changing
electric field, as described by the displacement
current such that 𝜵 × 𝑩 = 𝝁𝟎 𝑱 +
𝝏𝑬
𝝁𝟎 𝜺𝟎
𝝏𝒕
• Maxwell’s Equations are now complete
Discussion of Assignment