Electrostatic force due to spherical shell of charge
R
z
24 Rq
ddRaddq sin2
Q
2
2
2
sin
ddRkQkQdqdF
𝑟=𝑧 �̂� (field point)
𝑟 ′=𝑅�̂� (source point)⃗ =𝑧 �̂�−𝑅�̂�
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Fd⃗
cosdF
Electrostatic force due to spherical shell of charge
R
z
Fd⃗
cosdF
Electrostatic force due to spherical shell of charge
R
z
Fd⃗
sindF
sindF
Only z-component survives
cosdFdFz
Electrostatic force due to spherical shell of charge
R
z
Rcos
rRz coscos
Why are we doing this?
To change variable to which is the polar angle
Electrostatic force due to spherical shell of charge
R
z
Rcos
coscos Rz
cossincos 2
2 RzddRkQdFdFz
cossin2
2 RzddRkQdFz
and are NOT independent
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
Law of cosines
Electrostatic force due to spherical shell of charge
R
z
cos2222 RzzR
zRzR2
cos222
zRdd22sin
cossin2
2 RzddRkQdFz
)cos(11sin 22 RzddRkQdFz
zzRzd
zRdRkQdFz 2
11 222
22
2
222
2
dzzRzd
zRkQdFz
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
222
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
2222
0
'
2
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
22222
0
'
22
dzzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
2
2222
0
'
2
dzRzd
zRkQdFz
Electrostatic force due to spherical shell of charge
R
z
dRzd
zRkQdFz 2
2222
02'
2
Electrostatic force due to spherical shell of charge
R
z
dRzdzRkQdFz 1
2 2
222
02'
Electrostatic force due to spherical shell of charge
R
z
dRzzRkQdFz 12
2 2
22
2'
Electrostatic force due to spherical shell of charge
R
z
Electrostatic force due to spherical shell of charge
R
z
= z - R
Electrostatic force due to spherical shell of charge
R
z
= z + R
Electrostatic force due to spherical shell of charge
dRzzRkQF
Rz
Rzz 12
22
2
Rz
Rz
zRz
zRkQF
22
2
RzRzRz
RzRzRz
zRkQFz
2222
2
RzRzRzRzzRkQFz
2
RzRzRzRzzRkQFz
2
RzRzRzRzzRkQFz
2
RzRkQFz 42
2
24zRkQFz
2zkQqFz
Electrostatic force due to spherical shell of charge (inside the shell)
R
z
= R - z
Electrostatic force due to spherical shell of charge (inside the shell)
R z
= R + z
dRzzRkQF
zR
zRz 12
22
2
zR
zR
zRz
zRkQF
22
2
zRRzzR
zRRzzR
zRkQFz
2222
2
RzzRRzzRzRkQFz
2
RzzRRzRzzRkQFz
2
Electrostatic force due to spherical shell of charge (inside the shell)
0zF
Electrostatic force due to spherical shell of charge (inside the shell)