ASSIGNMENT 4 Course: PHYS2016 Due date: Thursday August 24th at 9am Hand in at the workshop or submit online. Late assignments will receive a mark of zero. Please attach a coversheet (available on Wattle). Do all questions Question 1. (15 marks) Consider a circular disk of radius a that lies in the xy plane and surface charge density

𝜎 𝑠′,𝜙′ =𝛼 + 𝛽𝑠′! sin𝜙′+ cos𝜙′

where (s′, φ′) are the usual cylindrical coordinate and A is a constant. In last weeks assignment you determined the electric field and potential at the point P located at (0, 0, z). This week calculate the potential at a point slightly displaced from the z axis at position r = zz +δ ss . You can obtain an expression by assuming that δs is small compared with all other length scales. Can you determine the electric field at P using this expression? Question 2. (15 marks) In the region a ≤ r ≤ b, a hollow spherical shell carries charge density (Figure 1)

Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r > b. Find the potential at the center using infinity as you reference point.

Figure 1.

ρ =kr2

Additional information (May or may not be required)

sin!𝜙𝑑𝜙 = 0!!

!

cos!𝜙𝑑𝜙 = 0!!

!

cos!𝜙𝑑𝜙 = 𝜋!!

!

sin!𝜙𝑑𝜙 = 𝜋!!

!

sin𝜙cos𝜙𝑑𝜙 = 0!!

!

sin𝜙𝑑𝜙 = −cos𝜙 + 𝐶

cos𝜙𝑑𝜙 = sin𝜙 + 𝐶

𝜕𝜕𝜙 sin𝜙cos𝜙 = cos!𝜙 − sin!𝜙

1

𝑥! + 𝑦! !/!  dx =𝑥

𝑦! 𝑥! + 𝑦! !/! + 𝐶

𝑥𝑥! + 𝑦! !/!  dx = −

1𝑥! + 𝑦! !/! + 𝐶

𝑥!

𝑥! + 𝑦! !/!  dx = −𝑥

𝑥! + 𝑦!+ ln 𝑥 + 𝑥! + 𝑦! + 𝐶