Lecture 3

```Electrostatic force due to spherical shell of charge
R
r
z


x
y
Spherical coordinates
z
  0 to 


x
y
z
  0 to 
  0 to 2
r  0 to R

x
y
x

z

y
Electrostatic force due to spherical shell of charge
q

4R 2
r
R
z
q0
dq  dA  R2 sin dd
kq0 dq kq0R 2 sin dd
dF 

2
2
r
r
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
r
R


z
dF cos 

dF
Electrostatic force due to spherical shell of charge
r
R


z
dF cos 

dF
Electrostatic force due to spherical shell of charge
r
R
dF sin 


z
Only z-component survives
dFz  dF cos

dF sin  dF
Electrostatic force due to spherical shell of charge
r
R


z
Rcos
z  R cos 
cos  
r
Why are we doing this?
To change variable to  which is the polar angle
Electrostatic force due to spherical shell of charge
r
R


z
Rcos
z  R cos 
cos  
r
kq0R 2 sin dd z  R cos
dFz  dF cos 
2
r
r
kq0R sin dd z  R cos
dFz 
r2
r
2
r and  are NOT independent
Electrostatic force due to spherical shell of charge
r
R


z
Electrostatic force due to spherical shell of charge
r
R
r


z
Electrostatic force due to spherical shell of charge
r
R


z
Electrostatic force due to spherical shell of charge
r
R


z
Law of cosines
Electrostatic force due to spherical shell of charge
r
R


z
r 2  R 2  z 2  2Rz cos
R2  z 2  r 2
cos 
2 zR
2rdr
 sin d  
2 zR
kq0R sin dd z  R cos
dFz 
r2
r
2
1
1
dFz  kq 0R sin d 2 d ( z  R cos  )
r
r
2
2
2
2


1
R  z  r 
2 rdr 1

dFz  kq0R
d  z  
2
zR r
r 
2z

2
2
2

 R  z  r  dr
kq0R
 2
dFz 
d  z  
z
2z
 r
 
2
2
2

 R  z  r  dr
kq0R
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
2
2
2


kq0R
R  z  r  dr
 2
dFz 
d  z  
z
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
kq0R
dF 
z 0
'
z
2
  R 2  z 2  r 2  dr
 2
d  z  
2z
 r
 
Electrostatic force due to spherical shell of charge
r
R
z
kq0R
dF 

0
z
'
z
2
 2 z 2  R 2  z 2  r 2  dr
d 
 2
2z

r
Electrostatic force due to spherical shell of charge
r
R
z
kq0R
dF 

0
z
'
z
2
 z 2  R 2  r 2  dr
d 
 2
2z

r
Electrostatic force due to spherical shell of charge
r
R
z
kq0R
dF 
2 0
2z
'
z
2
 z 2  R2  r 2 
d 
 dr
2
r


Electrostatic force due to spherical shell of charge
r
R
z
kq0R
dF 
2 0
2z
'
z
2
 z 2  R2 
d 
 1 dr
2
 r

Electrostatic force due to spherical shell of charge
r
R
z
2
2


kq0R
z R
'
dFz 
2 
 1 dr
2
2
2z
 r

Electrostatic force due to spherical shell of charge
r
R
z
Electrostatic force due to spherical shell of charge
R
r
z
r=z-R
Electrostatic force due to spherical shell of charge
R
r
z
r=z+R
Electrostatic force due to spherical shell of charge
zR
 z 2  R2 
kq0R
Fz 
  
 1dr
2
2
z
r

zR 
kq0R   z  R
Fz 
 r  
2
z
  r
2
2
zR


 z  R
 z 2  R2  
 z 2  R 2  
kq0R 
   z  R  
 
Fz 
  z  R  
2


z
z

R
z

R


 

 
kq 0R
Fz 
 z  R  z  R   z  R   z  R 
2
z
kq 0R
Fz 
 z  R  z  R  z  R  z  R 
2
z
kq 0R
Fz 
 z  R  z  R  z  R  z  R 
2
z
kq 0R
Fz 
 4R
2
z
kq0 4R
Fz 
2
z
kq 0 q
Fz  2
z
2
Electrostatic force due to spherical shell of charge
(inside the shell)
R
r
z
r=R-z
Electrostatic force due to spherical shell of charge
(inside the shell)
r
R
z
r=R+z
Electrostatic force due to spherical shell of charge
(inside the shell)
R z
2
2
0
z
2
2
R z
R z
2
2
0
z
2
R z
kq R
F 

z

z  R

 1dr

 r

kq R   z  R
F 
 r  
z
  r



 z 2  R2  
 z 2  R 2  
kq0R 
   R  z  
 
Fz 
  R  z  
2


z
R

z
R

z


 

 
kq 0R
Fz 
 R  z  z  R   R  z  z  R 
2
z
kq 0R
Fz 
 z  R  z  R  R  z  z  R 
2
z
Electrostatic force due to spherical shell of charge
(inside the shell)
Fz  0
```