Download Lecture 13: Displacement Current

Lecture 13 : Displacement current
• Displacement current
• Maxwell’s complete equations
Recap (1)
• Maxwell’s Equations for the 𝐸𝐸-field and 𝐵𝐵-field generated by
stationary (i.e., not varying with time) charge density 𝜌𝜌 and
current density 𝐽𝐽⃗ are:
𝝆𝝆
𝜵𝜵. 𝑬𝑬 =
𝜺𝜺𝟎𝟎
𝜵𝜵. 𝑩𝑩 = 𝟎𝟎
𝜵𝜵 × 𝑬𝑬 = 𝟎𝟎
𝜵𝜵 × 𝑩𝑩 = 𝝁𝝁𝟎𝟎 𝑱𝑱⃗
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, 𝑉𝑉 = −
𝜕𝜕Φ
𝜕𝜕𝑡𝑡
• This phenomenon is known as
electromagnetic induction, and
leads to the equation
modification 𝜵𝜵 × 𝑬𝑬 +
𝝏𝝏𝑩𝑩
𝝏𝝏𝝏𝝏
= 𝟎𝟎
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 𝛻𝛻 × 𝐵𝐵 =
⃗ Since 𝛻𝛻. 𝛻𝛻 × 𝐵𝐵 = 0, we find 𝛻𝛻. 𝐽𝐽⃗ = 0
𝜇𝜇0 𝐽𝐽.
𝜕𝜕𝜕𝜕
• However, charge conservation implies that 𝛻𝛻. 𝐽𝐽⃗ + = 0
𝜕𝜕𝜕𝜕
• This contradiction implies that Ampere’s law must be wrong for
time-varying currents!
Displacement current
𝝏𝝏𝑬𝑬
⃗
• This is fixed by the modification 𝜵𝜵 × 𝑩𝑩 = 𝝁𝝁𝟎𝟎 𝑱𝑱 + 𝝁𝝁𝟎𝟎 𝜺𝜺𝟎𝟎
𝝏𝝏𝝏𝝏
• Now, take the divergence of both sides. We obtain 𝜇𝜇0 𝛻𝛻. 𝐽𝐽⃗ +
𝜕𝜕𝐸𝐸
𝜇𝜇0 𝜀𝜀0 𝛻𝛻. = 0. Cancelling 𝜇𝜇0 and
𝜕𝜕𝜕𝜕
𝜕𝜕
𝜕𝜕
, this becomes 𝛻𝛻. 𝐽𝐽⃗ + 𝜀𝜀0
𝛻𝛻. 𝐸𝐸
𝜕𝜕𝑡𝑡
𝜕𝜕𝜕𝜕
• Now substituting in 𝛻𝛻. 𝐸𝐸 =
𝜌𝜌
,
𝜀𝜀0
swapping the order of 𝛻𝛻 and
=0
𝜕𝜕𝜕𝜕
we find 𝛻𝛻. 𝐽𝐽⃗ + = 0
𝜕𝜕𝜕𝜕
• The new version of the equation is consistent with charge
conservation!
Displacement current
• For time-varying magnetic fields : 𝜵𝜵 × 𝑬𝑬
𝝏𝝏𝑩𝑩
+
𝝏𝝏𝝏𝝏
= 𝟎𝟎
𝝏𝝏𝑬𝑬
⃗
• For time-varying electric fields : 𝜵𝜵 × 𝑩𝑩 = 𝝁𝝁𝟎𝟎 𝑱𝑱 + 𝝁𝝁𝟎𝟎 𝜺𝜺𝟎𝟎
𝝏𝝏𝝏𝝏
• 𝐸𝐸- and 𝐵𝐵-fields are symmetric : a changing 𝑩𝑩-field generates
an 𝑬𝑬-field, and a changing 𝑬𝑬-field generates a 𝑩𝑩-field
𝜕𝜕𝐸𝐸
𝜀𝜀0
𝜕𝜕𝑡𝑡
• The term
in the 2nd equation is known as the
displacement current, since it acts like a current density and is
equal to
𝜕𝜕𝐷𝐷
,
𝜕𝜕𝑡𝑡
in terms of the electric displacement 𝐷𝐷 = 𝜀𝜀0 𝐸𝐸
Displacement current
• Second, consider a capacitor discharging into a circuit:
• Consider applying Ampere’s Law ∮ 𝐵𝐵. 𝑑𝑑 𝑙𝑙⃗ = 𝜇𝜇0 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 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 𝐸𝐸 = 𝜎𝜎/𝜀𝜀0 in terms
of the charge density 𝜎𝜎
• If 𝐴𝐴 is the plate area, then 𝑄𝑄 = 𝜎𝜎𝜎𝜎
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
hence 𝐼𝐼 =
= 𝜀𝜀0 𝐴𝐴
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
• This is equal to the displacement
current enclosed by the surface!
Maxwell’s complete equations
• Maxwell’s Equations are now complete!
• In a vacuum:
𝝆𝝆
𝜵𝜵. 𝑬𝑬 =
𝜺𝜺𝟎𝟎
𝜵𝜵. 𝑩𝑩 = 𝟎𝟎
𝝏𝝏𝑩𝑩
𝜵𝜵 × 𝑬𝑬 +
= 𝟎𝟎
𝝏𝝏𝝏𝝏
𝝏𝝏𝑬𝑬
𝜵𝜵 × 𝑩𝑩 − 𝝁𝝁𝟎𝟎 𝜺𝜺𝟎𝟎
= 𝝁𝝁𝟎𝟎 𝑱𝑱⃗
𝝏𝝏𝝏𝝏
Maxwell’s complete equations
• More generally, in materials with relative permittivity 𝜀𝜀𝑟𝑟 and
relative permeability 𝜇𝜇𝑟𝑟 we can use the electric displacement
𝐷𝐷 = 𝜀𝜀𝑟𝑟 𝜀𝜀0 𝐸𝐸 and magnetic intensity 𝐻𝐻 = 𝐵𝐵/𝜇𝜇𝑟𝑟 𝜇𝜇0 :
𝜵𝜵. 𝑫𝑫 = 𝝆𝝆
𝜵𝜵. 𝑩𝑩 = 𝟎𝟎
𝝏𝝏𝑩𝑩
𝜵𝜵 × 𝑬𝑬 +
= 𝟎𝟎
𝝏𝝏𝝏𝝏
𝝏𝝏𝑫𝑫
𝜵𝜵 × 𝑯𝑯 −
= 𝑱𝑱⃗
𝝏𝝏𝝏𝝏
Maxwell’s complete equations
• The equations can be expressed in either differential
form or integral form, and these are equivalent
Differential form
Integral form
𝛻𝛻. 𝐷𝐷 = 𝜌𝜌
� 𝐷𝐷. 𝑑𝑑𝐴𝐴⃗ = 𝑄𝑄𝑒𝑒𝑒𝑒𝑒𝑒
𝜕𝜕𝐵𝐵
=0
𝛻𝛻 × 𝐸𝐸 +
𝜕𝜕𝜕𝜕
𝛻𝛻. 𝐵𝐵 = 0
𝜕𝜕𝐷𝐷
𝛻𝛻 × 𝐻𝐻 −
= 𝐽𝐽⃗
𝜕𝜕𝜕𝜕
� 𝐸𝐸. 𝑑𝑑𝑙𝑙⃗ = −
𝜕𝜕
� 𝐵𝐵. 𝑑𝑑𝐴𝐴⃗
𝜕𝜕𝜕𝜕
� 𝐵𝐵. 𝑑𝑑𝐴𝐴⃗ = 0
� 𝐻𝐻. 𝑑𝑑𝑙𝑙⃗ = 𝐼𝐼𝑒𝑒𝑒𝑒𝑒𝑒 +
𝜕𝜕
� 𝐷𝐷. 𝑑𝑑 𝐴𝐴⃗
𝜕𝜕𝜕𝜕
Name or effect
Gauss’s Law
Electromagnetic
induction
No monopoles
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
𝝏𝝏𝑬𝑬
⃗
current such that 𝜵𝜵 × 𝑩𝑩 = 𝝁𝝁𝟎𝟎 𝑱𝑱 + 𝝁𝝁𝟎𝟎 𝜺𝜺𝟎𝟎
• Maxwell’s Equations are now complete
𝝏𝝏𝝏𝝏
Discussion of Assignment