Electrostatic force due to spherical shell of charge

transcript

R

z

24 Rq

ddRaddq sin2

Q

2

sin

ddRkQkQdqdF

𝑟=𝑧 �̂� (field point)

𝑟 ′=𝑅�̂� (source point)⃗ =𝑧 �̂�−𝑅�̂�

R

z

R

z

R

z

R

z

R

z

R

z

R

z

R

z

R

z

R

z

R

z

R

z

Fd⃗

cosdF

R

z

Fd⃗

cosdF

R

z

Fd⃗

sindF

Only z-component survives

cosdFdFz

R

z

Rcos

rRz coscos

Why are we doing this?

To change variable to which is the polar angle

R

z

Rcos

coscos Rz

cossincos 2

2 RzddRkQdFdFz

cossin2

2 RzddRkQdFz

and are NOT independent

R

z

R

z

R

z

R

z

Law of cosines

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

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

222

2

dzzRzd

zRkQdFz

R

z

2

2222

0

'

2

dzzRzd

zRkQdFz

R

z

2

22222

0

'

22

dzzRzd

zRkQdFz

R

z

2

2222

0

'

2

dzRzd

zRkQdFz

R

z

dRzd

zRkQdFz 2

2222

02'

2

R

z

dRzdzRkQdFz 1

2 2

222

02'

R

z

dRzzRkQdFz 12

2 2

22

2'

R

z

R

z

= z - R

R

z

= z + R

dRzzRkQF

Rz

Rzz 12

22

2

Rz

zRz

zRkQF

22

2

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

R z

= R + z

dRzzRkQF

zR

zRz 12

22

2

zR

zRz

zRkQF

22

2

zRRzzR

zRkQFz

2222

2

RzzRRzzRzRkQFz

2

RzzRRzRzzRkQFz

2

0zF

Electrostatic force due to spherical shell of charge

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