Ampres Law: Magnetic Field Inside a Wire
Consider a long, straight wire of radius R. The current is I distributed uniformly over the cross section. I Apply Ampres law, B d = 0 IC , to the circular loop of radius r < R. The symmetry dictates that the magne
Some important Symbols
(based on http:/icl.pku.edu.cn/yujs/MathWorld/math/k/k137.htm and http:/icl.pku.edu.cn/yujs/MathWorld/math/p/p267.htm)
Kronecker Delta symbol is dened by dened by
i j = ei ej ,
where i, j, k are indices denoting three perpendicular
Legendre Polynomials and their properties
Original by Gregory E. Amenta, Modied by A. Gangopadhyaya March 5, 2006
Let us consider Laplaces equation,
2
V =0
(1)
in spherical coordinates (r, , ): 1 r2 r r2 V r + 1 2 sin r sin V + 1 r2 sin2 2V 2 = 0. (2)
At
Legendre Polynomials
http:/plasma.physics.swarthmore.edu/Physics112/old_assignments.html
Legendre polynomials are a complete set of orthogonal functions in spherical coordinates. This means that any function can be represented as a sum of Legendre polynom
LAPLACES EQUATION IN SPHERICAL COORDINATES With Applications to Electrodynamics We have seen that Laplaces equation is one of the most significant equations in physics. It is the solution to problems in a wide variety of fields including thermodynamics an
1. Is it correct to say that product of Answer: and A?
A is a vector eld that is perpendicular to A because it is a cross
Answer is NO! For example, one can show
A is not necessarily perpendicular
to the vector A, as one would expect from a cross produc
Image Charge for a grounded sphere (radius R) with a point charge q sitting at (0, 0, d < R).
Consider a point charge q sitting at (0, 0, d) inside a sphere of radius R (d < R). Assume the image charge q situated at (0, 0, d ) outside the sphere. The pote
Location and value of an image charge for grounded conducting sphere
Consider a grounded sphere of radius R with a charge q at (0, 0, a), where a > R. We want to determine potential at every point outside the sphere. We want to simulate this situation by
Formulae
Vsph (r, , ) =
l=0
Al rl +
Bl rl+1
Pl (cos )
Vcyl (, ) = (a0 + b0 log ) (c0 + d0 ) +
n=1
an n + bn n (cn cos(n) + dn sin(n)
(E2 E1 ) n12 =
a
0
; (E2 E1 ) n12 = 0 ;
a
cos(a + b) = cos a cos b sin a sin b
a
sin
0
nx a mx sin dx = mn ; a a 2
cos
0
Dr. Gangopadhyaya
Final Exam
Physics 351
April 25, 2008
As always, to receive full credit you must show your work in a neat, organized and a legible form. Some Relevant Formuae are given below:
Vsph (r, , ) =
l=0
Al rl +
Bl rl+1
Pl (cos )
(1)
Vcyl (, ) =
A Brief Outline of the History of Electromagnetism
Richard Alan Peters II April 5, 2000
The primary sources for this outline were From Falling Bodies to Radio Waves, Emilio Segr`, Freeman, New York, 1984, A History of Electricity and Mage netism, Herbert
Physics 351
Electricity and Magnetism Prof. Asim Gangopadhyaya
Test1
I will not divulge any information about this exam to any one for any reason. Name 1) The integral a da
S
is sometimes called the vector area of the surface S. Use problem 60a to show th
A property of solutions of Laplaces equation This chapter mostly deals with solutions of Laplaces equation. Electrostatic potential obeys Laplaces equation at point with no charge density. There is a very special property of solutions of Laplaces equation
Chapter 2
How would you write down the volume charge densities of following objects: A ring of radius R carries a total charge Q (uniformly distributed) placed on the xy -plane with its center at the origin. ans. (r) = Q (r R) ( ) . 2R2 2
A ring of radiu
Divergence and Curl in a nutshell
As an undergraduate student, I found Divergence and Curl operators to be quite confusing. That is why I am providing this graphical presentation to you. to you. Hope this helps.
Divergence:
Divergence measures the rate at
Dierentiating under integral sign
Asim Gangopadhyaya1
Department of Physics, Loyola University Chicago, 6525 N. Sheridan Rd., Chicago IL 60626.
1 e-mail:
[email protected]
1
In Supersymmetric quantum mechanics one often calculates a quantity
x2
E W 2 (x) dx
Curl, Laplacian and all that!
Asim Gangopadhyaya
We all have seen a problem where a river has a dierential velocity on its surface. The velocity is parallel to the shore everywhere and it has maximum magnitude at the center and symmetrically goes down on
Dr. Gangopadhyaya
Sample Tests I
Physics 351
February 19, 2006
Questions from one of you First question: On of the sample tests (TST1 351 F01.pdf), you asked to determine the divergence of the electric eld in the form (q/4 P i epsilon) (r a)/abs(r a)3 , w
The history of science is science itself; the history of the individual, the individual. Johann Wolfgang von Goethe Mineralogy and Geology Mathematics, rightly viewed, possesses not only truth, but supreme beauty-a beauty cold and austere, like that of sc
Dr. Gangopadhyaya
Test III
Physics 351
December 4, 2002
As always, to receive full credit you must show your work in a neat, organized and a legible form. Some Relevant Formuae are given below:
Vsph (r, , ) =
l=0
Al r l +
Bl rl+1
Pl (cos )
Vcyl (, ) = (a0
No Name on this page - only on the back. Thanks!
Physics 351 Elec. & Magnetism Test 2 Prof. Asim Gangopadhyaya As always, to receive full credit you must show your work in a neat, organized and a legible form. Some Relevant Formulae are given below:
Vsph
Dr. Gangopadhyaya
Test II
Physics 351
May 8, 2005
As always, to receive full credit you must show your work in a neat, organized and a legible form. Some Relevant Formuae are given below:
Vsph (r, , ) =
l=0
Al rl +
Bl rl+1
Pl (cos )
(1)
Vcyl (, ) = (a0 +
Test 1
Physics 351
March 6, 2005
Dr. Gangopadhyaya: As always, to receive full credit you must show your work in a neat, organized and a legible form. Getting an integral from the calculator will not suce. Binomial Expansion : For x < 1, (1 + x)n = 1 + n
Test I
Dr. Gangopadhyaya
Physics 351
Oct. 12, 98
As always, to receive full credit you must show your work in a neat, organized and a legible form. Partial credit will be given only for well deserving work. Do not write answer to more than one problems in
Test I
Physics 351/Dr. Gangopadhyaya
February 22, 2005
To receive full credit your work needs to be a) correct, and b) neat. Do not write answer to more than one problems in one sheet. When done, put problems in ascending order and staple all pages.
1) El
Dr. Gangopadhyaya
Test I
Physics 351
February
20, 2006
To receive full credit you must show your work in a neat, organized and a legible form. First five problems are worth 14 points each and the last problem is worth 30 points. Einstein summation convent
Dr. Gangopadhyaya
Extra Credit 3
Physics 351
May 5, 2006
1. At the surface of the earth the magnetic field is approximately point dipole at the center of the earth having a dipole moment: mE where mE
the same as the field from a
= mE
[i sin eo cos</>o +
3
Physics 351
Electricity and Magnetism Prof. Asim Gangopadhyaya
Test 2
A cubical cavity sits with one end at the origin. It has no charge inside it. The potentials on the planar surfaces of the cavity are given by: V (0, y, z ) = V0 , V (, y, z ) = 0, V (x