Maxwell’s 1st and 2nd Equationscblake/Class4_Maxwells... · •Maxwell’s 1st Equation expresses...
Transcript of Maxwell’s 1st and 2nd Equationscblake/Class4_Maxwells... · •Maxwell’s 1st Equation expresses...
![Page 1: Maxwell’s 1st and 2nd Equationscblake/Class4_Maxwells... · •Maxwell’s 1st Equation expresses Gauss’s Law in the differential form 𝜵.𝑬=𝝆 𝜺𝟎 (field lines end](https://reader034.fdocuments.in/reader034/viewer/2022042406/5f20bae25d57b81c88498a4d/html5/thumbnails/1.jpg)
Class 4 : Maxwell’s Equations for Electrostatics
• Concept of charge density
• Maxwell’s 1st and 2nd Equations
• Physical interpretation of divergence and curl
• How do we check whether a given vector field could be an electric field?
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Recap
𝑆
• The electric field 𝐸 around a charge distribution can be determined by
Gauss’s Law 𝐸. 𝑑 𝐴 = 𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑/휀0
• The 𝐸-field may be described as the gradient of an electrostatic potential
𝑉, where 𝐸 = −𝛻𝑉
• The potential difference is the work done in moving a unit charge
between 2 points, ∆𝑉 = − 𝐸. 𝑑 𝑙
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• We often want to merge point charges together, and think of a continuous distribution of charge
• We define the charge density at position 𝑥, or charge per unit volume, as 𝜌( 𝑥)
𝑥
• Consider a volume element 𝛿𝑉
• Charge in this volume element 𝛿𝑄 = 𝜌 𝛿𝑉
• Total charge 𝑄 = 𝜌 𝛿𝑉
𝛿𝑉
Origin
Charge density
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Vector divergence
• We can apply 𝛻 =𝜕
𝜕𝑥,𝜕
𝜕𝑦,𝜕
𝜕𝑧to a vector function 𝐸 =
(𝐸𝑥, 𝐸𝑦, 𝐸𝑧) using either a dot product or a cross product
• We will first consider the dot product. Using the usual dot
product rule, we have 𝛻.𝐸 =𝜕𝐸𝑥
𝜕𝑥+𝜕𝐸𝑦
𝜕𝑦+𝜕𝐸𝑧
𝜕𝑧
• This is known as the divergence of a vector field (we will explain why shortly!)
• We will meet the cross product version soon!
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Maxwell’s 1st Equation
• A powerful result of vector calculus, which links a surface integral with the divergence, is the vector divergence theorem
• If 𝑆 is a closed surface around a volume 𝑉, then for any vector
field 𝐸 we can say:
𝑉
𝑆 𝐸. 𝑑 𝐴 = 𝛻. 𝐸 𝑑𝑉
Divergence theorem
𝐸
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Maxwell’s 1st Equation
• The divergence of a vector field at a point 𝑃 measures how much it is flowing out of, or into, that point
• We can see this from 𝐸. 𝑑 𝐴 = 𝛻.𝐸 𝑑𝑉 - the divergence
𝛻.𝐸 creates an outward flux 𝐸.𝑑 𝐴
Physical meaning of divergence
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Consider the following vector field. What is the divergence in the box?
A. Zero everywhere
B. Non-zero everywhere
C. Zero some places, non-zero other places
D. Impossible to tell
Zero
eve
ryw
here
Non-zer
o ever
ywher
e
Zero
som
e pla
ces,
non-zer
o ...
Imposs
ible
to te
ll
0% 0%0%0%
Clicker question
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Consider the following vector field. What is the divergence in the box?
A. Zero everywhere
B. Non-zero everywhere
C. Zero some places, non-zero other places
D. Impossible to tell
Zero
eve
ryw
here
Non-zer
o ever
ywher
e
Zero
som
e pla
ces,
non-zer
o ...
Imposs
ible
to te
ll
0% 0%0%0%
Clicker question
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Maxwell’s 1st Equation
• Apply the vector divergence theorem to Gauss’s Law:
• Re-arranging: 𝛻. 𝐸 −𝜌
𝜀0𝑑𝑉 = 0 (for any volume)
• This implies: 𝜵. 𝑬 =𝝆
𝜺𝟎
𝐸 .𝑑 𝐴 =𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑휀0
𝛻.𝐸 𝑑𝑉 𝜌 𝑑𝑉
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Maxwell’s 1st Equation
• Maxwell’s beautiful equation 𝛻. 𝐸 =𝜌
𝜀0is describing
mathematically how electric field lines flow away from positive charges and toward negative charges
Please note in workbook
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Maxwell’s 1st Equation
• Consider the following electric field:
• What is the corresponding charge density and electrostatic potential?
𝑥
𝐸𝑥𝐸0
−𝐸0
𝑎−𝑎
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Vector curl
• We have already met the divergence of a vector field, 𝛻.𝐸 =𝜕𝐸𝑥
𝜕𝑥+𝜕𝐸𝑦
𝜕𝑦+𝜕𝐸𝑧
𝜕𝑧where 𝛻 = (
𝜕
𝜕𝑥,𝜕
𝜕𝑦,𝜕
𝜕𝑧)
• We now consider applying the 𝛻 operator to a vector using
the cross product ×, which is known as a vector curl 𝛻 × 𝐸
• The curl is evaluated as a cross product and is a vector, e.g.
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Type equation here.• A fundamental result in vector calculus called Stokes’ Theorem
relates the vector curl to a line integral
• If 𝑆 is any surface bounded by a closed loop 𝐿, then for any
vector field 𝐸 we can say:
𝐸. 𝑑 𝑙 = (𝛻 × 𝐸). 𝑑 𝐴
Integral around closed loop
Stokes’ Theorem
Integral over surface
𝐸 𝑑 𝐴
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Physical meaning of vector curl
• The divergence (𝛻. ) describes the amount of outflow or inflow of a vector field at a point
• The curl (𝛻 ×) describes the circulation of a vector field about a point (and the axis of the circulation)
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Physical meaning of vector curl
• The divergence (𝛻. ) describes the amount of outflow or inflow of a vector field at a point
• The curl (𝛻 ×) describes the circulation of a vector field about a point (and the axis of the circulation)
• We can see this is the case using Stokes’ Theorem : if
𝛻 × 𝐸 > 0, then there will be a non-zero integral around
a closed loop, 𝐸. 𝑑 𝑙 > 0, corresponding to a circulation
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Consider the vector field shown. What can we say about its curl?
A. Zero everywhere
B. Non-zero everywhere
C. Zero in some places and non-zero in other places
D. Impossible to tell
Zero
eve
ryw
here
Non-zer
o ever
ywher
e
Zero
in so
me
place
s and n
on...
Imposs
ible
to te
ll
0% 0%0%0%
Clicker question
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Maxwell’s 2nd equation
• We can use the above results to deduce Maxwell’s 2nd
equation (in electrostatics)
• If we move an electric charge in a closed loop we will do zero
work : 𝐸. 𝑑 𝑙 = 0
• Using Stokes’ Theorem, this implies that for any surface in an
electrostatic field, 𝛻 × 𝐸 . 𝑑 𝐴 = 0
• Since this is true for any surface, then 𝜵 × 𝑬 = 𝟎
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Maxwell’s 2nd equation
• We could have deduced the same result using the fact that
the electric field can be expressed as a potential, 𝐸 = −𝛻𝑉
• Applying the curl operator to both sides, we find 𝛻 × 𝐸 =
− 𝛻 × 𝛻𝑉 = 0 (which can be shown by multiplying out the
components of 𝛻 × 𝛻𝑉)
• 𝜵 × 𝑬 = 𝟎 is Maxwell’s 2nd equation as applied to electrostatics
• (It must be modified for time-varying fields as we will describe in later lectures)
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Maxwell’s 2nd equation
• In physical terms, the fact that 𝛻 × 𝐸 = 0 is describing that electric field lines do not circulate into loops – they always look like diagram (a) below, not diagram (b)
Please note in workbook
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Which of the following could be an electrostatic field in the
region shown?
A. Both
B. Only I
C. Only II
D. Neither
Both
Only I
Only II
Neith
er
0% 0%0%0%
Clicker question
I II
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Summary
• Vector calculus is a powerful mathematical tool based
on the vector operator 𝛻 =𝜕
𝜕𝑥,𝜕
𝜕𝑦,𝜕
𝜕𝑧
• Maxwell’s 1st Equation expresses Gauss’s Law in the
differential form 𝜵. 𝑬 =𝝆
𝜺𝟎(field lines end on charges)
• Maxwell’s 2nd Equation for electrostatics is 𝜵 × 𝑬 = 𝟎(field lines do not circulate)
• This is equivalent to defining an electrostatic potential
𝑉 as 𝑬 = −𝜵𝑽