Data Loading...
Curvilinear coordinates - NISER Flipbook PDF
Curvilinear coordinates Curvilinear Rectangular Spherical Polar Cylindrical coordinates coordinates x, y, z r, θ, φ ρ, φ
112 Views
22 Downloads
FLIP PDF 87.25KB
Curvilinear coordinates Curvilinear coordinates coordinates range
transformation line element d~l surface element d~s volume element dτ
Rectangular
Spherical Polar
Cylindrical
x, y, z −∞ ≤ x ≤ ∞ −∞ ≤ y ≤ ∞ −∞ ≤ z ≤ ∞ – – – ˆ ˆidx + ˆjdy + kdz n ˆ dydz dxdydz
r, θ, φ r≥0 0≤θ≤π 0 ≤ φ ≤ 2π x = r sin θ cos φ y = r sin θ sin φ z = r cos θ ˆ ˆ sin θdφ rˆdr + θrdθ + φr 2 rˆr sin θdθdφ r2 sin θdθdφdr
ρ, φ, z ρ≥0 0 ≤ φ ≤ 2π −∞ ≤ z ≤ ∞ x = ρ cos φ y = ρ sin φ z=z ˆ ρˆdρ + φρdφ + zˆdz ρˆρdφdz ρdρdφdz
Gradient: ˆi ∂ + ˆj ∂ + kˆ ∂ φ ∂x ∂y ∂z ∂ 1 ∂ 1 ∂ ˆ ˆ ∇φ(r, θ, φ) = rˆ + θ +φ φ ∂r r ∂θ r sin θ ∂φ ∂ 1 ∂ ∂ ˆ φ ∇φ(ρ, φ, z) = ρˆ + φ + zˆ ∂ρ ρ ∂φ ∂z ∇φ(x, y, z) =
For divergence and curl in different coordinate system, please refer to Griffith’s front cover. Example 10. Find volume and surface area of (a) a sphere of radius R, (b) a cylinder of radius R and length h. Example 11. Find (a) gradient of 1/r, (b) gradient of f (~r), (c) curl of ~r/r3 . (r 6= 0)
1
Introduction to electrostatics and magnetostatics Electromagnetic interaction is the most prevalent kind of interaction we encounter in daily life. Apart from gravity, every other sort like electricity, mobile phone, friction, surface tension, chemical reaction and bonding, molecular interaction, cell functioning, DNA structure and thousand other things are based on electromagnetic interaction. Naturally it makes sense to study electromagnetism in some details, both in the macroscopic (i.e. classical) and sub-atomic (i.e. quantum) level. A good starting point of the study is electromagnetism in simple setting – (a) when the electric charges are static and electric field is not changing with time and (b) when we have steady flow of electric charges and static magnetic field. The first case is called electrostatics, while the second magnetostatics. A brief history -500: Greek philosophers observed something that attracted very light objects and named them elektron, while those attracting iron only were named magnesia. 1730: C.F. du Fay proposed two types of charge: positive and negative. 1786: Priestly, Cavendish and Coulomb came up with inverse square law of electrostatics interaction. 1820: Oersted and Ampere demonstrated relation between electricity and magnetism by deflecting magnetic needle with electric current. 1831: Faraday discovered magnetic induction. 1873: Maxwell theoretically explain all electric and magnetic phenomena observed till date through Maxwell’s equations. 1905: Einstein proposed special theory of relativity demanding invariance of Maxwell’s equations under boost. Relative strength of electromagnetic interaction fundamental interaction strong E&M Weak Gravity
carrier of interaction gluon photon W ±, Z 0 graviton
relative strength 1037 1032 1024 1
distance scale 10−10 m ∞ 10−13 m ∞
A few salient features • Electric and magnetic field can travel in vacuum. Ether as medium of propagation was ruled out by Einstein’s STR and Michelson-Morley experiment. • Electric charge is quantized and conserved. • Magnetic charge or monopole has not yet been found, its existance is debated. • Electric field is stronger than mgnetic field by a factor of c, the velocity of light. 2
Electrostatics: Coulomb’s law All the source charges are stationary, but the test charges may move. Make sure the test charges are weak enough not to distrub the original source charge distribution. Coulomb’s law: The force on a test charge Q due to a single point charge q which is at rest a distance r away is 1 qQ rˆ (1) F~ = 4π0 r2 where 0 = 8.85 × 10−12 C2 /N − m2 is called the permitivity of free space and ~r = ~rQ − ~rq , ~rQ and ~rq being position vectors for field and source coordinates respectively. The force F~ is repulsive if q and Q have the same sign and attractive if their signs are opposite. Also, according to Newton’s third law F~q = −F~Q . Superposition principle: The force on test charge Q due to all other charges is equal to the vector sum of forces due to individual charges and completely unaffected by presence of other charges, X F~m , (2) F~ = F~1 + F~2 + . . . ≡ m
which for a set discrete charges {q1 , q2 , . . .} can be written as F~ =
1 X qm Q rˆm 2 4π0 m rm
(3)
where rm is the distance of the test charge Q from qm source charge. Coulomb’s law and superposition principle constitute physical input for electrostatics. Electric / Electrostatic field: To avoid the confusion related to “Action at distance” problem, the idea of electric field is introduced. A working definition of elestrostatic field is force of source charge(s) on unit test charge Q = 1, ~ ≡ E
1 X qm ~ rˆ ⇒ F~ = QE. 2 m 4π0 m rm
(4)
~ r) is a vector field function of ~r. Physically it is rather difficult The electrostatic field E(~ concept to understand, but can be conveniently viewed as some kind of stress the space, medium or vacuum, surrounding the source charge(s) experience. It can be visualized with ~ converges on “-” and diverges from “+” the help of field lines by tracing out direction of E, ~ while the magnitude of E is denoted by density of field lines (denser the lines, stronger the ~ = 0. field). The field lines never cross each other, if so that is where E Continuous charge distributions: In all our previous definition of electrostatic force and field we have assumed single or assembly of point discrete charges. If instead the charge is distributed continuously over some region, • take the limit qm → dq = ρdτ, σda, λdl R P • convert sum to volume or surface or line integrals as appropriate m → R R R R P • substitute m qm → V ρdτ or S σda or C λdl, where ρ, σ, λ are volume, surface and line charge density resp. 3
• The volume, surface etc. integrals are over the source coordinates that is where the charges are. Thus the electric fields for volume, surface and line distribution of charge are, Z 1 ρ(~r0 ) ~ E(~r) = rˆdτ 0 volume charge density 4π0 V r2 Z 1 σ(~r0 ) ~ r) = E(~ rˆda0 surface charge density 4π0 S r2 Z λ(~r0 ) 1 ~ rˆdl0 line charge density E(~r) = 4π0 C r2
4
(5) (6) (7)