Electric Charges and Fields

An arbitrary surface encloses a dipole. What is the electric flux through this surface?  

Total charge -Q is uniformly spread along the length of a ring of radius R. A small test charge +q of mass m is kept at the centre of the ring and is given a gentle push along the axis of the ring.
(a) Show that the particle executes a simple harmonic oscillation.
(b) Obtain its time period.

Two charges –q each are fixed separated by distance 2d. A third charge q of mass m placed at the mid-point is displaced slightly by x (x<<d) perpendicular="" to="" the="" line="" joining="" two="" fixed="" charged="" as="" shown="" in="" fig.="" 1.14.="" show="" that="" q="" will="" perform="" simple="" harmonic="" oscillation="" of="" time="" period.<="" strong=""></d)>

T=[8π³ɛ_o md³/q²]^1/2 

There is another useful system of units, besides the SI/mks A system, called the cgs (centimeter-gram-second) system. In this system Coloumb’s law is given by F=Qq/r2 vector r where the distance r is measured in cm (= 10–2 m), F in dynes (=10–5 N) and the charges in electrostatic units (es units), where 1es unit of charge=1/[3]x10-9C The number [3] actually arises from the speed of light in vaccum which is now taken to be exactly given by c = 2.99792458 × 108m/s. An approximate value of c then is c = [3] × 108 m/s. (i) Show that the coloumb law in cgs units yields 1 esu of charge = 1 (dyne)1/2 cm. Obtain the dimensions of units of charge in terms of mass M, length L and time T. Show that it is given in terms of fractional powers of M and L. (ii) Write 1 esu of charge = x C, where x is a dimensionless number. Show that this gives 

Two fixed, identical conducting plates (α &β ) , each of surface area S are charged to –Q and q, respectively, where Q > q > 0. A third identical plate (γ ), free to move is located on the other side of the plate with charge q at a distance d . The third plate is released and collides with the plate β . Assume the collision is elastic and the time of collision is sufficient to redistribute charge amongst β &γ .  (a) Find the electric field acting on the plate γ before collision.   (b) Find the charges on β and γ after the collision. 

Consider a sphere of radius R with charge density distributed as ρ (r ) = kr for r ≤ R for r > R .

(a) Find the electric field at all points r. 

(b) Suppose the total charge on the sphere is 2e where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.

In 1959 Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density N, which is maintained a constant. Let the charge on the proton be: ep=−(1+y)e where e is the electronic charge. Find the critical value of y such that expansion may start

Five charges,q each are placed at the comers of a regular pentagon of side a.          

i) What will be the electric field at O, the centre of the pentagon?
(ii) What will be the electric field at O if the charge from one of the comers (say A) is removed?
(iii) What will be the electric field at O if the charge q at A is replaced by – q?
(b) How would your answer to (a) be affected if pentagon is replaced by n-sided regular polygon with charge q at each of its comers?

1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocity  = 2.0 × 10⁶ m s^–1. If E between the plates separated by 0.5 cm is 9.1 × 10² N/C, where will the electron strike the upper plate? (|e|=1.6 × 10^–19 C, m_e = 9.1 × 10^–31 kg.)