Electric Field of charged hollow and solid Spheres
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1 lectic Field of chaged hollow and solid Sphees Fits F.M. de Mul -field of a chaged hollow o solid sphee 1
2 esentations: lectomagnetism: Histoy lectomagnetism: lect. topics lectomagnetism: Magn. topics lectomagnetism: Waves topics Capacito filling (complete) Capacito filling (patial) Divegence Theoem -field of a thin long chaged wie -field of a chaged disk -field of a dipole -field of a line of dipoles -field of a chaged sphee -field of a polaied obect -field: field enegy lectomagnetism: integations lectomagnetism: integation elements Gauss Law fo a cylindical chage Gauss Law fo a chaged plane Laplace s and oisson s Law B-field of a thin long wie caying a cuent B-field of a conducting chaged sphee B-field of a homogeneously chaged sphee Downloads: scoll to lecticity and Magnetism -field of a chaged hollow o solid sphee
![[C/m ], o volume chage density ρ [C/m 3 ] σ](/docs-images/89/98264716/images/3-3.jpg "and ρ can be f (position) Question:")
3 -field of a chaged sphee Available: A sphee, adius, with suface chage density σ [C/m ], o volume chage density ρ [C/m 3 ] σ and ρ can be f (position) Question: Calculate -field in abitay points inside and outside the sphee -field of a chaged hollow o solid sphee 3
4 -field of a chaged sphee Contents: 1. A hollow sphee, homogeneously chaged (conducting). Idem., non-homogeneously chaged 3. A solid sphee, homogeneously chaged 4. Idem., non-homogeneously chaged Method: integation ove chage elements (adial and angula integations). NB. Homogeneously chaged sphees can also be calculated in an easy way using the symmety in Gauss Law. -field of a chaged hollow o solid sphee 4
5 -field of a chaged sphee Contents: 1. A hollow sphee, homogeneously chaged (conducting). Idem., non-homogeneously chaged 3. A solid sphee, homogeneously chaged 4. Idem., non-homogeneously chaged -field of a chaged hollow o solid sphee 5
6 1. Hollow sphee, homogeneously chaged Z 1. Chage distibution: (suface chage) s [C/m ]. Coodinate axes: e e Z-axis = pola axis 3. Symmety: spheical 4. Spheical coodinates:,, -field of a chaged hollow o solid sphee 6
7 1. Hollow sphee, homogeneously chaged x i. Z y Q i 1. Z-axes. oint on Z-axis 3. all Q i s at i, i, i contibute i to in 4. In : i,x, i,y, i, 5. xpect fom symmety: S ( i,x + i,y ) = 6. = e only! -field of a chaged hollow o solid sphee 7
8 1. Hollow sphee, homogeneously chaged d. Z d Distibuted chage: dq d dq 4 ( e e ) d dq at da chage element dq = s da.sin e suface element da=(.d).(.sin.d) d and (.e ): see next page -field of a chaged hollow o solid sphee 8
9 1. Hollow sphee, homogeneously chaged d d. Coss section though O:.sin.sin Z e d dq at da e -.cos.cos d.sin =(.sin) + ( -.cos) (.e )= ( -.cos) / -field of a chaged hollow o solid sphee 9
10 1. Hollow sphee, homogeneously chaged d d. Z e d dq d s da 4 ( e da=(.d).(.sin.d) =(.sin) + ( -.cos) (.e )= ( -.cos) / e ).sin These expessions can be used fo both and < d d s. d..sin. d.[.cos ] 4 3/ [(.sin ) (.cos ) ] s.sin.[.cos ] d d 3/ -field of a chaged hollow 4 o solid [( sphee.sin ) (.cos 1 ) ]
11 1. Hollow sphee, homogeneously chaged.sin d d. Z e d d d dq s.sin.[.cos ] d 3/ 4 [(.sin ) (.cos ) ] Integation: ove φ : esults in facto π ove θ : eplace sinθ.dθ by -d(cosθ), divide above and below by 3, call cosθ = x, and / = a and a +1-ax = y (with -a.dx = dy) and integate ove y esult of integation: s F y, with F y a y a cases: a > 1 and a < 1. This expession can also be used fo othe limiting values of θ. -field of a chaged hollow o solid sphee 11
12 1. Hollow sphee, homogeneously chaged.sin d d. Z e d d dq s.sin.[.cos ] d 3/ 4 [(.sin ) (.cos ) ] esult of integation: s F y, with F y a y a with a= / and y = a +1-a.cos θ. cases: a > 1 and a < 1. This expession can also be used fo othe limiting values of θ. d Next slide: plot of F fo vaying values of the integation limits : begin θ b =..18 ; end θ e =18 θ b = : closed sphee θ b > : bowl shape -field of a chaged hollow o solid sphee 1
13 1. Hollow sphee, homogeneously chaged s F y, with F y a y a 1st cuve, /a^ : s s a = accoding to Gauss F() a = : inside sphee: = /a^ a = () / thb=3 thb=9 thb=15 with a= / and y = a +1-a.cos θ. thb= thb=6 thb=1 thb=18 lot of F fo vaying values of the integation limits : begin θ b =..18 ; end θ e =18 θ b = : closed sphee θ b > : bowl shape In plot: θ b = thb -field of a chaged hollow o solid sphee 13 θ b θ e =18 bowl
14 1. Hollow sphee, homogeneously chaged s F y, with F y a y a 1st cuve, /a^ (= Gauss ) (abs.value) F() Coespondence: fo thb= with esult fom Gauss. esult = inside sphee thb=9 : Symmetic aound = fo half sphee thb=3 and 15 : thb=6 and 1 : mutual symm. aound = inside sphee a = : inside sphee: = /a^ a = () / thb=3 thb=9 thb=15 thb= thb=6 thb=1 thb=18 θ b bowl -field of a chaged hollow o solid sphee 14 - θ e =18
15 1. Hollow sphee, homogeneously chaged s F y, with F with a= / and y = a +1-a.cos θ. y a y a a = : inside sphee: = F().6.4. lot of F fo integation limits : begin θ b = 45; end θ e =135 (ing) θ b -4-4 a = () / θ e ing -.6 -field of a chaged hollow o solid sphee 15
16 .sin 1. Hollow sphee, homogeneously chaged d Z e. d d d d dq s.sin.[.cos ] d 3/ 4 [(.sin ) (.cos ) ] esult fo in : < : = > : s Q e e 4 using σ = Q / (4π ). With Gauss: Gauss sphee at adius : Q. da encl 4 -field of a chaged hollow o solid sphee 16 Q
17 -field of a chaged sphee Contents: 1. A hollow sphee, homogeneously chaged (conducting). Idem., non-homogeneously chaged 3. A solid sphee, homogeneously chaged 4. Idem., non-homogeneously chaged -field of a chaged hollow o solid sphee 17
![. Hollow sphee, non-homogeneously chaged Now σ = f (θ,φ).sin d x d Z e d. d y d s.sin.[.cos ] d 3/ 4 [(.sin ) (.](/docs-images/89/98264716/images/18-2.jpg "cos ) ] and -components may be pesent; e x and e y ae e d d dq.sin.sin -field of a chaged hollow o solid sphee 18 e =(.sin) + ( -.cos) -.cos.cos Fo : (.")
18 . Hollow sphee, non-homogeneously chaged Now σ = f (θ,φ).sin d x d Z e d. d y d s.sin.[.cos ] d 3/ 4 [(.sin ) (.cos ) ] and -components may be pesent; e x and e y ae e d d dq.sin.sin -field of a chaged hollow o solid sphee 18 e =(.sin) + ( -.cos) -.cos.cos Fo : (.e )= ( -.cos) / Fo x and y we need:.sin /
![. Hollow sphee, non-homogeneously chaged Now σ = f (θ,φ) d s.sin.[.cos ] d 3/ 4 [(.sin ) (.](/docs-images/89/98264716/images/19-1.jpg "cos ) ].sin d d. d d y x Z e d d dq x, y d Fo x and y we need:.sin / s.sin. d 4 [(.")
19 . Hollow sphee, non-homogeneously chaged Now σ = f (θ,φ) d s.sin.[.cos ] d 3/ 4 [(.sin ) (.cos ) ].sin d d. d d y x Z e d d dq x, y d Fo x and y we need:.sin / s.sin. d 4 [(.sin ) ( with [.sin. ( ).cos ) ] ( ) cos (fo -field of a chaged hollow o solid sphee 19 sin (fo In many situations, e.g. asymmetic chage distibutions, numeical integation will be necessay. x y ); ). 3/
20 -field of a chaged sphee Contents: 1. A hollow sphee, homogeneously chaged (conducting). Idem., non-homogeneously chaged 3. A solid sphee, homogeneously chaged 4. Idem., non-homogeneously chaged -field of a chaged hollow o solid sphee
21 3. Solid sphee, homogeneously chaged.sin d d. Z e d d d dq d Deived fo a hollow sphee : s.sin.[.cos ] d 3/ 4 [(.sin ) (.cos ) ] Fo a solid sphee we need an exta integation vaiable: ove vaying adius. Suppose: adius vaies fom to. Integation element d Suface chage element dq = σ. da = σ..dθ..sinθ.dφ (σ in C/m ) has to beplaced by Volume chage element: (ρ in C/m 3 ) dq= ρ. dv= ρ. d..dθ..sinθ.dφ.sin.[.cos ] d d d 3/ 4 -field of a chaged hollow o solid [(.sin ) (.cos sphee 1 ) ]
22 3. Solid sphee, homogeneously chaged 3/ ] ).cos ( ).sin [( ].cos.[.sin 4 d d d Z. dq d d d e.sin d Integation ove θ and φ esulted in: with a= / and y = a +1-a.cos θ. s with, a y y a y F F d a y y a y F df d with d,. d d 3 3 : d d -field of a chaged hollow o solid sphee ) do not contibute! (sphee shells 3 3 : 3 d esult fo a hollow sphee:
23 3 3. Solid sphee, homogeneously chaged Z. dq d d d e.sin d esult fo a solid sphee: 3 3 : d And with Q =ρ. (4/3). π 3 : 4. : : inside 4 : : outside 3 Q Q These expession ae accoding to Gauss. d -field of a chaged hollow o solid sphee : d invese uadatic linea
24 -field of a chaged sphee Contents: 1. A hollow sphee, homogeneously chaged (conducting). Idem., non-homogeneously chaged 3. A solid sphee, homogeneously chaged 4. Idem., non-homogeneously chaged -field of a chaged hollow o solid sphee 4
25 .sin 4. Solid sphee, non-homogeneously chaged d x d Z e. d d y x, y d.sin.[.cos ] d d d 3/ 4 [(.sin ) (.cos ) ] d dq ρ = f (,θ,φ) : - and -components pesent. d d.sin. d 4 [(.sin ) ( with [.sin ( ) cos (fo sin (fo. ( ).cos ) ] x y ); ). 3/ d -field of a chaged hollow o solid sphee In geneal: Integation has to be pefomed numeically. 5
26 Conclusion fo homogeneous chage distibution: = > : < : hollow: solid: Hollow: Q tot = σ.4π Q tot 4 Q tot. 4 e e Solid: Q tot = ρ.4π 3 /3 total chage seems to be in cente the end -field of a chaged hollow o solid sphee 6
Complete Solution to Potential and E-Field of a sphere of radius R and a charge density ρ[r] = CC r 2 and r n
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Complete Solution to Potential and E-Field of a sphee of adius R and a chage density ρ[] = CC 2 and n Deive the electic field and electic potential both inside and outside of a sphee of adius R with a
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