Transcript Physics 321 Hour 10 Coordinate Systems and Multiparticle Systems

Physics 321
Hour 10
Coordinate Systems and Multiparticle
Systems
Central Force
If a vector field is radial and depends only on r, what
is its curl?
1 𝜕𝐹𝑧 𝜕𝐹𝜃
𝜕𝐹𝑟 𝜕𝐹𝑧
1 𝜕𝑟𝐹𝜃 𝜕𝐹𝑟
𝛻×𝐹 =𝑟
−
+𝜃
−
+𝑧
−
𝑟 𝜕𝜃
𝜕𝑧
𝜕𝑧
𝜕𝑟
𝑟 𝜕𝑟
𝜕𝜃
𝛻×𝐹
1
1 𝜕 sin 𝜃 𝐹𝜑 𝜕𝐹𝜃
1 𝜕𝐹𝑟 1 𝜕𝑟𝐹𝜑
=𝑟
−
+𝜃
−
𝑟 sin 𝜃 𝑟
𝜕𝜃
𝜕𝜑
𝑟 sin 𝜃 𝜕𝜑 𝑟 𝜕𝑟
1 𝜕𝑟𝐹𝜃 𝜕𝐹𝑟
+𝜑
−
𝑟 𝜕𝑟
𝜕𝜃
Notes on Coordinates
A vector can be expressed as
𝐹 = 𝐹𝑥 𝑥 + 𝐹𝑦 𝑦 + 𝐹𝑧 𝑧
or as
𝐹 = 𝐹𝑟 𝑒𝑟 + 𝐹𝜗 𝑒𝜗 + 𝐹𝜑 𝑒𝜑
𝐴 ∙ 𝐵 = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧
Is the following expression also valid?
𝐴 ∙ 𝐵 = 𝐴𝑟 𝐵𝑟 + 𝐴𝜗 𝐵𝜗 + 𝐴𝜑 𝐵𝜑
If and only if 𝐴 and 𝐵 are defined at the same point.
A Potential Energy for 1 Charge
𝑘𝑒 𝑞1 𝑞2
𝑈 𝑟 =
+ 𝑚𝑔𝑦
𝑟
𝜕𝑈
𝜕𝑈
𝜕𝑈
𝐹 = −𝛻𝑈 = −
𝑥−
𝑦−
𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑟=
𝑥2 + 𝑦2 + 𝑧2
𝜕𝑈 𝜕𝑈 𝜕𝑟
𝑘𝑒 𝑞1 𝑞2 1
𝑘𝑒 𝑞1 𝑞2 𝑥
=
=−
2𝑥 = −
2
𝜕𝑥
𝜕𝑟 𝜕𝑥
𝑟
2𝑟
𝑟3
A Potential Energy for 3 Charges
𝑈 𝑟1 , 𝑟2 , 𝑟3
𝑘𝑒 𝑞1 𝑞2
𝑘𝑒 𝑞2 𝑞3
𝑘𝑒 𝑞3 𝑞1
=
+
+
+ 𝑚1 𝑔𝑦1
|𝑟2 − 𝑟1 | |𝑟3 − 𝑟2 | |𝑟1 − 𝑟3 |
+ 𝑚2 𝑔𝑦2 + 𝑚3 𝑔𝑦3 .
𝜕𝑈
𝜕𝑈
𝜕𝑈
𝐹1 = −𝛻1 𝑈 = −
𝑥−
𝑦−
𝑧
𝜕𝑥1
𝜕𝑦1
𝜕𝑧1
𝜕𝑈
𝜕𝑈
𝜕|𝑟2 − 𝑟1 |
𝜕𝑈
𝜕|𝑟1 − 𝑟3 |
=
+
𝜕𝑥1 𝜕|𝑟2 − 𝑟1 | 𝜕𝑥1
𝜕|𝑟1 − 𝑟3 | 𝜕𝑥1
𝑟2 − 𝑟1
=
(𝑥2 − 𝑥1 )2 +(𝑦2 − 𝑦1 )2 +(𝑧2 − 𝑧1 )2