Download Topic 1: Vectors

Lecture 9 : Maxwell’s Equations for 𝐵
• Determining magnetic fields
• Maxwell’s Equations for 𝐵
• Magnetic vector potential
Recap (1)
• Magnetic forces are a
fundamental phenomenon of
nature generated by electric
currents
• We say that a current (or magnet)
sets up a magnetic field 𝐵 around
it, which causes another current
(or magnet) to feel a force
Recap (2)
• The Biot-Savart Law and Ampere’s Law are two equivalent
methods for describing how magnetic fields 𝐵 are generated
by a current 𝐼
Ampere’s Law for 𝐵
Biot-Savart Law for 𝐵
𝑑𝐵
𝑃
𝑟
𝐼
𝜇0 𝐼 𝑑 𝑙 × 𝑟
𝑑𝐵 =
4𝜋 𝑟 2
𝐼
𝑑𝑙
𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐
Determining magnetic fields
• We now introduce the current loop (or solenoid),
which is equivalent to a bar magnet
Similar 𝐵-field lines
Determining magnetic fields
• What is the 𝐵 field at the centre of a current loop?
𝐼
𝑟
𝐵
• Biot-Savart Law : 𝑑𝐵 =
𝜇0 𝐼𝑑 𝑙 × 𝑟
4𝜋 𝑟 2
• The contribution 𝑑𝐵 from all
current elements reinforce
𝜇
𝐼
• Hence 𝐵 = 𝑑𝐵 = 0 2 𝑑𝑙
4𝜋 𝑟
where 𝑑𝑙 = 2𝜋𝑟
• We find:
𝐵=
𝜇𝑜 𝐼
2𝑟
Determining magnetic fields
• Now, consider a solenoid of length 𝑙 with 𝑁 turns
carrying current 𝐼. What is the 𝐵-field in the middle?
Loop for Ampere’s
Law
• Apply Ampere’s Law
𝐵 . 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 to the
closed loop shown
•
𝐵. 𝑑 𝑙 = 𝐵 × 𝑙,
neglecting 𝐵 outside coil
• 𝐼𝑒𝑛𝑐 = 𝑁 × 𝐼
• Hence
𝐵=
𝜇0 𝑁 𝐼
𝑙
Maxwell’s Equations for 𝐵
• We have already met Maxwell’s two equations for
𝝆
electrostatics: 𝜵. 𝑬 = and 𝜵 × 𝑬 = 𝟎. What are
𝜺𝟎
the equivalent relations for the magnetic field 𝐵?
• Take the divergence (𝛻.) of the Biot-Savart law:
𝛻. 𝑑𝐵 = 𝛻.
𝜇0 𝐼 𝑑 𝑙×𝑟
4𝜋 𝑟 2
=
(where we have used the relation 𝛻
𝜇0 𝐼
𝑑𝑙. 𝛻
4𝜋
1
𝑟
=
×𝛻
1
𝑟
=0
𝑟
)
𝑟2
• We deduce the fundamental result for any current
distribution, 𝜵. 𝑩 = 𝟎
Maxwell’s Equations for 𝐵
• If 𝛻. 𝐵 = 0 everywhere, what does this mean
physically?
• Divergence theorem implies 𝐵. 𝑑 𝐴 = 0 : there is no
inward/outward flux of magnetic field from any point
Maxwell’s Equations for 𝐵
• Hence there are no magnetic charges (“monopoles”) and
magnetic field lines never end and always form closed loops
Maxwell’s Equations for 𝐵
• The final Maxwell’s Equation uses a fundamental result in
vector calculus known as Stokes’ Theorem
• If 𝑆 is any surface bounded by a closed loop 𝐿, then for any
vector field 𝐵 we can say:
𝐵. 𝑑 𝑙 =
Integral around
closed loop
(𝛻 × 𝐵). 𝑑 𝐴
Integral over
surface
Maxwell’s Equations for 𝐵
• But from Ampere’s Law, we have
𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 = 𝜇0 𝐽. 𝑑𝐴
where 𝐽 is the current density of the enclosed current 𝐼𝑒𝑛𝑐
• We hence deduce:
(𝛻 × 𝐵). 𝑑 𝐴 = 𝜇0
𝐽. 𝑑 𝐴
𝜵 × 𝑩 = 𝝁𝟎 𝑱
• This describes how magnetic fields 𝐵 are produced by current 𝐽
Maxwell’s Equations for 𝐵
• If 𝛻 × 𝐵 = 𝜇0 𝐽 everywhere, what does this mean physically?
• The curl (𝛻 ×) of a vector field describes its tendency to
circulate around a point, so this relation is saying that
magnetic fields circulate around currents
Maxwell’s Equations for 𝐵
• We have now deduced all four Maxwell’s Equations for the
electric field 𝐸 and magnetic field 𝐵 generated by stationary
charge density 𝜌 and current density 𝐽:
𝝆
𝜵. 𝑬 =
𝜺𝟎
𝜵×𝑬=𝟎
𝜵. 𝑩 = 𝟎
𝜵 × 𝑩 = 𝝁𝟎 𝑱
• In later lectures, we will discuss how these equations must be
modified to account for time-varying situations
Maxwell’s Equations for 𝐵
• The equations can be expressed in either differential
form or integral form, and these are equivalent
Differential form
𝜌
𝛻. 𝐸 =
𝜀0
𝛻×𝐸 =0
𝛻. 𝐵 = 0
𝛻 × 𝐵 = 𝜇0 𝐽
Integral form
Name or effect
𝑄𝑒𝑛𝑐
𝐸. 𝑑𝐴 =
𝜀0
Gauss’s Law
(Coulomb’s Law)
𝐸. 𝑑𝑙 = 0
Electrostatic potential
𝐸 = −𝛻𝑉
𝐵. 𝑑𝐴 = 0
No monopoles, magnetic
potential 𝐵 = 𝛻 × 𝐴
𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐
Ampere’s Law
(Biot-Savart Law)
Magnetic vector potential
• In electrostatics, we saw that the electric field 𝐸 could be
generated from a scalar field 𝑉(𝑟) called the electrostatic
potential, such that 𝑬 = −𝜵𝑽
• This relation always satisfies 𝛻 × 𝐸 = 0, and the zero-point
of 𝑉 is arbitrary (i.e., an arbitrary constant can be added)
• The magnetic field 𝐵 can likewise be generated from a
magnetic potential, however this is a vector field 𝐴(𝑟) such
that 𝑩 = 𝜵 × 𝑨
• This relation always satisfies 𝛻. 𝐵 = 0, since 𝛻. (𝛻 × 𝐴) = 0
Magnetic vector potential
• The magnetic vector potential 𝐴 also has a freedom in its
definition: we can add an arbitrary constant to 𝛻. 𝐴
• We choose to add a constant such that 𝜵. 𝑨 = 𝟎
• (More advanced comment:) To see why, we need to use a relation
from vector calculus that comes from applying two curls! That relation
is 𝛻 × 𝛻 × 𝐴 = 𝛻 𝛻. 𝐴 − 𝛻. 𝛻 𝐴
• Substituting in 𝛻 × 𝐵 = 𝜇0 𝐽 we find 𝛻 2 𝐴 − 𝛻 𝛻. 𝐴 = −𝜇0 𝐽
• Note that if a constant is added to 𝛻. 𝐴, the gradient 𝛻(𝛻. 𝐴) does not
change, hence the generation of 𝐴 from currents 𝐽 is not affected
Magnetic vector potential
• There are hence analogous relations between the
generation of electrostatic potential 𝑉 from charge density
𝜌, and magnetic vector potential 𝐴 from current density 𝐽
• These are useful for higher physics development
Electrostatic potential 𝑉:
Magnetic potential 𝐴:
𝜌
𝛻 𝑉=−
𝜀0
𝛻 2 𝐴 = −𝜇0 𝐽
2
Summary
• The magnetic field 𝐵 satisfies
the fundamental relation
𝜵. 𝑩 = 𝟎. In physical terms,
𝐵-field lines form closed loops
• The generation of the 𝐵-field
from currents is described by
𝜵 × 𝑩 = 𝝁𝟎 𝑱, completing
Maxwell’s Equations
• The magnetic field can be
generated from a vector
potential 𝐴, via 𝐵 = 𝛻 × 𝐴