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(phys223)[2009](f)midterm~ma_yxf^_10544.pdf
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PHYS223
Intermediate Electricity and Magnetism I
Midterm Examination
18/Oct/2009
Time allowed: 3 hours

You may find the following formulae useful:

Coulombs law:
. E field due to a point charge 2014.q...Err


. E field due to a set of discrete charges 21014.niiiiq.....Err


. E field due to a line P with linear charge density ....r



....2014.Pdl.......rErrr

. E field due to a surface S with surface charge density ....r



....2014.Sda.......rErrr

. E field due to a volume . with volume charge density ....r



....2014.d........rErrr


Curl E in electrostatics: 0Cd.......E0El



Gausss law of E:
. Differential form

. For a point charge q at : .r
..30q.......Err


. For a volume charge density .: 0.....E



. Integral form enc001SQdd.........Ea




Electric potential:
. Integral form ..Vd....rOrEl


. Differential form V...E




Poissons equation of V: 20V.....

Laplaces equation of V: 20V..


Boundary conditions of E and V across surfaces:
. ..abovebelowabovebelow00.VVnn...............EEn


. ////abovebelowabovebelowVV...EE



where is a unit vector perpendicular to the plane and pointing from below to above .n







1. (a) Show that the vector field
, .......,,xyzAyxz....vxyz

where A is a constant, cannot be an electric field in electrostatics.

(b) The potential inside a region is given in spherical coordinates by
, ..2cos,,VrAr....

where A is a constant. Find the electric field at all points where . 0r.


(c) The electric field inside a region is given in cylindrical coordinates by
....22..sincosfor 0..sincosfor ARsRAsRs.................sEs...

where A is a constant. Find the charge distribution.


2. An insulating sphere with radius R carries a spherically symmetric volume charge density given by
, ..2rAr....

where A is a constant.
The total amount of charge carried by the sphere is Q.

(a) Express A in terms of Q and R.
(b) Find the potential at the center of the sphere in terms of Q and R. Take infinity as the reference point.










3. A semicircular ring with radius R lies on the xy plane and is centered at the origin, as shown in the figure below. The ring carries a uniform linear charge density ..


z





r



D







R y
d.


.
.r


dl.





x

Consider the observation point r on the positive z-axis at a distance D from the center of the ring. The field at this point is the sum of all contributions due to all small length elements . dl.


Consider a small source at position , at which the angle between the positive x-axis and the line joining the point to the origin is .. The length subtends an angle d. at the centre, as shown in the above figure. .r
dl.


(a) Express in terms of