phy103a: lecture # 4home.iitk.ac.in/~akjha/phy103_notes_hw_solutions/phy103_lec_4.โฆย ยท โข dirac...
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Semester II, 2017-18 Department of Physics, IIT Kanpur
PHY103A: Lecture # 4
(Text Book: Intro to Electrodynamics by Griffiths, 3rd Ed.)
Anand Kumar Jha 10-Jan-2018
2
Notes โข The Solutions to HW # 1 have been posted.
โข Course Webpage: http://home.iitk.ac.in/~akjha/PHY103.htm
โข Office Hour โ Friday 2:30-3:30 pm
โข HW # 2 has also been posted.
โข I am assuming that everyone has been receiving the course emails.
โข The fundamental Theorem for derivative:
โข The fundamental Theorem for Gradient: ๏ฟฝ ๐ป๐ป๐๐ โ ๐๐๐ฅ๐ฅ = ๐๐ ๐๐ โ ๐๐(๐๐)
๐๐
๐๐ ๐๐๐๐๐๐๐
โข The fundamental Theorem for Divergence
๏ฟฝ ๐ป๐ป โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข
๐๐๐๐๐๐
๏ฟฝ ๐ป๐ป ร ๐๐ โ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐ฅ๐ฅ
๐๐๐๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข โข The fundamental Theorem
for Curl
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๏ฟฝ๐๐๐๐๐๐๐๐
๐๐๐๐ = ๐๐ ๐๐ โ ๐๐(๐๐) ๐๐
๐๐
(Gaussโs theorem):
(Stokesโ theorem):
Summary of Lecture # 3:
โข Line integral, surface integral, and volume integral
โข Coordinate systems: Spherical polar, and cylindrical
Dirac Delta function:
โข Dirac delta function is a special function, which is defined as:
๐ฟ๐ฟ ๐๐ = 0, ๐๐๐๐ ๐๐ โ 0 = โ, ๐๐๐๐ ๐๐ = 0 ๏ฟฝ ๐ฟ๐ฟ ๐๐ ๐๐๐๐ = 1
โ
โโ
๐ฟ๐ฟ ๐๐
๐๐
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๐ฟ๐ฟ ๐๐ = lim๐ ๐ โ0
12๐๐๐ ๐ 2
exp โ๐๐ โ ๐๐ 2
2๐ ๐ 2
โข Realization of a Dirac Delta function
โข Example: What is the charge density of a point charge ๐๐ kept at the origin?
๐๐(๐๐) = ๐๐๐ฟ๐ฟ(๐๐); ๏ฟฝ ๐๐ ๐๐ ๐๐๐๐ =โ
โโ๏ฟฝ ๐๐๐ฟ๐ฟ ๐๐ ๐๐๐๐ = ๐๐โ
โโ
Properties of a Dirac Delta function:
(4) 3D Dirac delta function is defined as: ๐ฟ๐ฟ3 ๐๐ = ๐ฟ๐ฟ ๐๐ ๐ฟ๐ฟ ๐ฆ๐ฆ ๐ฟ๐ฟ ๐ง๐ง ๏ฟฝ ๏ฟฝ ๏ฟฝ ๐๐ ๐๐ ๐ฟ๐ฟ3 ๐๐ โ ๐๐
โ
โโ
โ
โโ= ๐๐(๐๐)
โ
โโ
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๐ฟ๐ฟ ๐๐ โ ๐๐ = 0, ๐๐๐๐ ๐๐ โ 0 = โ, ๐๐๐๐ ๐๐ = ๐๐
(3) If ๐๐(๐๐) is a continuous function of ๐๐
๏ฟฝ ๐ฟ๐ฟ ๐๐ โ ๐๐ ๐๐๐๐ = 1โ
โโ
๏ฟฝ ๐๐(๐๐)๐ฟ๐ฟ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐(๐๐)๐ฟ๐ฟ ๐๐ โ ๐๐ ๐๐๐๐ = ๐๐(๐๐)โ
โโ
โ
โโ
๐ฟ๐ฟ ๐๐๐๐ =1๐๐๐ฟ๐ฟ(๐๐), (1)
(2) Dirac delta function centered at ๐๐ = ๐๐ is defined as follows
Divergence of the vector field ๐๐ = r๏ฟฝr2
Recall Homework Problem 1.5. The divergence ๐๐ โ r๏ฟฝrn = 2โ๐๐
r(n+1).
So, for ๐๐ = r๏ฟฝr2 , ๐๐ โ ๐๐ = ๐๐ โ r๏ฟฝ
r2 = 0r3 = 0 (except at r=0 where it is 0/0, not defined
Letโs calculate the divergence using the divergence theorem:
๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข
๐๐๐๐๐๐
๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข= ๏ฟฝ
rฬR2 โ ๐ ๐ 2sin๐๐ ๐๐๐๐ ๐๐๐๐rฬ = ๏ฟฝ sin๐๐ ๐๐๐๐ ๐๐๐๐ = 4๐๐
Take the volume integral over a sphere of radius R and the surface integral over the surface of a sphere of radius R.
Therefore,
๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข= 4๐๐
๐๐๐๐๐๐
We find that ๐๐ โ ๐๐=0 but its integral over a volume is finite. This is possible only if
๐๐ โ ๐๐ = ๐๐ โ rฬr2
= 4๐๐ ๐ฟ๐ฟ(๐ซ๐ซ)
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Divergence of the vector field ๐๐ = r๏ฟฝr2
Recall Homework Problem 1.5. The divergence ๐๐ โ r๏ฟฝrn = 2โ๐๐
r(n+1).
So, for ๐๐ = r๏ฟฝr2 , ๐๐ โ ๐๐ = ๐๐ โ r๏ฟฝ
r2 = 0r3 = 0 (except at r=0 where it is 0/0, not defined)
Letโs calculate the divergence using the divergence theorem:
๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข
๐๐๐๐๐๐
๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข= ๏ฟฝ
rฬR2 โ ๐ ๐ 2sin๐๐ ๐๐๐๐ ๐๐๐๐rฬ = ๏ฟฝ sin๐๐ ๐๐๐๐ ๐๐๐๐ = 4๐๐
Take the volume integral over a sphere of radius R and the surface integral over the surface of a sphere of radius R.
Therefore,
๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข= 4๐๐
๐๐๐๐๐๐
We find that ๐๐ โ ๐๐=0 but its integral over a volume is finite. This is only possible if
๐๐ โ ๐๐ = ๐๐ โ rฬr2
= 4๐๐ ๐ฟ๐ฟ(๐ซ๐ซ)
7
๐๐ โ rฬr2
= 4๐๐ ๐ฟ๐ฟ(r) = 4๐๐ ๐ฟ๐ฟ(๐ซ๐ซ โ ๐ซ๐ซ๐ซ)
๐๐ โ rฬr2
? ?
A few more essential concepts:
(i) The divergence ๐๐ โ ๐ ๐ is known (ii) The curl ๐๐ ร ๐ ๐ is known (iii) If the field goes to zero at infinity.
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1. The Helmholtz theorem
For the proof of this theorem, see Appendix B of Griffiths
A vector field ๐ ๐ in elctrodynamics can be completely determined if:
2. Scalar Potential:
(1) โซ ๐ ๐ โ ๐๐๐ฅ๐ฅb
a is independent of path. (2) โฎ๐ ๐ โ ๐๐๐ฅ๐ฅ = 0 for any closed loop.
This is because of Stokesโ theorem
๏ฟฝ ๐๐ ร ๐ ๐ โ ๐๐๐๐ = ๏ฟฝ ๐ ๐ โ ๐๐๐ฅ๐ฅ
๐๐๐๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข
โข This is because Curl of a gradient is zero ๐๐ ร ๐๐V = ๐๐
(3) ๐ ๐ is the gradient of a scalar function: ๐ ๐ = โ๐๐V
If the curl of a vector field ๐ ๐ is zero, that is, if ๐๐ ร ๐ ๐ = 0 everywhere, then:
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โข The scalar potential not unique. A constant can be added to V without affecting the gradient: ๐๐V= ๐๐(V + a), since the gradient of a constant is zero.
โข The minus sign is purely conventional.
A few more essential concepts:
3. Vector Potential:
(1) โซ๐ ๐ โ ๐๐๐๐ is independent of surface. (2) โฎ๐ ๐ โ ๐๐๐๐ = 0 for any closed surface.
This is because of the divergence theorem
๏ฟฝ ๐๐ โ ๐ ๐ ๐๐๐๐ = ๏ฟฝ ๐ ๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข
๐๐๐๐๐๐
โข This is because divergence of a curl is zero ๐๐ โ ๐๐ ร ๐๐ = 0
If the divergence of a vector field ๐ ๐ is zero, that is, if ๐๐ โ ๐ ๐ = 0 everywhere, then:
(3) ๐ ๐ is the gradient of a vector function: ๐ ๐ = ๐๐ ร ๐๐
10
โข The vector potential is not unique. A gradient ๐๐V of a scalar function can be added to ๐๐ without affecting the curl, since the curl of a gradient is zero.
A few more essential concepts:
Electric Field
Force on a test charge ๐๐ due to a single point charge ๐๐ is:
๐ ๐ =1
4๐๐๐๐0 ๐๐ ๐๐r2
rฬ
๐๐0 = 8.85 ร 10โ12C2
N โ m2 Permittivity of free space
Force on a test charge Q due to a collection of point charges is:
๐ ๐ ๐ซ๐ซ =1
4๐๐๐๐0 ๐๐1 ๐๐r1
2 r1 ๏ฟฝ +๐๐2 ๐๐r2
2 r2 ๏ฟฝ +
๐๐3 ๐๐r3
2 r3 ๏ฟฝ + โฏ
(in the units of Newton)
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0 ๐๐1r1
2 r1 ๏ฟฝ +๐๐2r2
2 r2 ๏ฟฝ +
๐๐3r3
2 r3 ๏ฟฝ + โฏ
Q: What is field (a vector function), physically? A: We donโt really know. At this level, field is just a mathematical concept which is consistent with the physical theory (electrodynamics). Also, we know how to calculate a field. 11
Coulombโs Law
= ๐๐๐๐(๐ซ๐ซ)
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๐๐r2
rฬ
Electric field due to a single point charge ๐๐ is:
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๏ฟฝ๐๐๐๐r2
rฬ
For a line charge ๐๐๐๐ = ๐๐ ๐ซ๐ซโฒ ๐๐๐๐ For a surface charge ๐๐๐๐ = ๐๐ ๐ซ๐ซโฒ ๐๐๐๐ For a volume charge ๐๐๐๐ = ๐๐ ๐ซ๐ซโฒ ๐๐๐๐
We know how to calculate electric fields due a charge distribution, using Coulombโs law. Weโll now explore some tricks for calculating the field more efficiently. 12
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0 ๐๐1r1
2 r1 ๏ฟฝ +๐๐2r2
2 r2 ๏ฟฝ +
๐๐3r3
2 r3 ๏ฟฝ + โฏ
Electric field due to a collection of point charges is:
Electric Field
Electric field due to a continuous charge distribution is:
Electric Flux and Gaussโs Law:
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๐๐r2
rฬ
Electric field ๐๐(๐ซ๐ซ) due to a single point charge ๐๐ at origin is:
13
Electric field is a vector field. One way to represent a vector field is by drawing a vectors of given magnitude and directions.
(i) Field lines emanate from the positive charge and end
up on the negative charge or go up to infinity.
Another way to represent a vector field is by drawing the field lines:
(ii) The density is the filed lines is proportional to the strength of the field.
The electric flux is defined as
ฮฆ๐ธ๐ธ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข
๐๐ โ ๐๐๐๐ is proportional the number of field lines passing through an area element ๐๐๐๐ When the area ๐๐๐๐ is perpendicular to the field ๐๐ the dot product is zero.
14
Electric Flux and Gaussโs Law:
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๐๐r2
rฬ
Electric field ๐๐(๐ซ๐ซ) due to a single point charge ๐๐ at origin is:
The electric flux due to a point charge ๐๐ at origin through a spherical shell of radius R.
ฮฆ๐ธ๐ธ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข= ๏ฟฝ ๏ฟฝ
14๐๐๐๐0
๐๐๐ ๐ 2
rฬ2๐๐
๐๐=0
๐๐
๐๐=0โ ๐ ๐ 2sin๐๐๐๐๐๐๐๐๐๐rฬ =
๐๐๐๐0
15
Electric Flux and Gaussโs Law:
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๐๐r2
rฬ
Electric field ๐๐(๐ซ๐ซ) due to a single point charge ๐๐ at origin is:
The electric flux due to a point charge ๐๐ at origin through any enclosing closed surface is
ฮฆ๐ธ๐ธ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=๐๐๐๐0
16
Electric Flux and Gaussโs Law:
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๐๐r2
rฬ
Electric field ๐๐(๐ซ๐ซ) due to a single point charge ๐๐ at origin is:
๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=๐๐enc๐๐0
This is the Gaussโs law in the integral form: The flux through a surface is equal to the total charge enclosed by the surface divided by ๐๐0
17
The electric flux due to a collection of point charges through any enclosing closed surface.
= ๏ฟฝ
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข๏ฟฝ๐๐๐๐
๐๐
๐๐=๐๐
.๐๐๐๐ = ๏ฟฝ ๐๐
๐๐=๐๐
๏ฟฝ
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข๐๐๐๐.๐๐๐๐ = ๏ฟฝ
๐๐๐๐๐๐0
๐๐
๐๐=๐๐
Electric Flux and Gaussโs Law:
๐๐(๐ซ๐ซ) =1
4๐๐๐๐0๐๐r2
rฬ
Electric field ๐๐(๐ซ๐ซ) due to a single point charge ๐๐ at origin is:
ฮฆ๐ธ๐ธ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=? ?
๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=๐๐enc๐๐0
Using divergence theorem
๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ ๐๐ โ ๐๐๐๐
๐๐๐ข๐ข๐ข๐ข๐ข๐ข
๐๐๐๐๐๐ ๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ =
๐๐enc๐๐0
๐๐๐๐๐๐
๐๐enc = ๏ฟฝ๐๐ ๐ซ๐ซโฒ ๐๐๐๐
๏ฟฝ ๐๐ โ ๐๐ ๐๐๐๐ = ๏ฟฝ
1๐๐0๐๐ ๐ซ๐ซโฒ ๐๐๐๐
๐๐๐๐๐๐
For a volume Charge density
๐๐ โ ๐๐ =๐๐ ๐๐0
This is the Gaussโs law in differential form
18
Or,
Or,
Electric Flux and Gaussโs Law:
Gaussโs law doesnโt have any information that the Coulombโs does not contain. The importance of Gaussโs law is that it makes calculating electric field much simpler and provide a deeper understanding of the field itself.
This is the Gaussโs law in integral form.
๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=๐๐enc๐๐0
This is the Gaussโs law in integral form.
Q: (Griffiths: Ex 2.10): What is the flux through the shaded face of the cube due to the charge ๐๐ at the corner
Answer:
๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=
124
๐๐๐๐0
19
Electric Flux and Gaussโs Law:
๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข ? ?
24๏ฟฝ ๐๐ โ ๐๐๐๐
๐ ๐ ๐ข๐ข๐ข๐ข๐ข๐ข=๐๐๐๐0