Q1.26Multiple choice
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.
This question refers to a figure in the original PDF.
- (a)Find the critical value of y such that expansion may start.
- (b)Show that the velocity of expansion is proportional to the distance from the centre. 1.27 Consider a sphere of radius R with charge density distributed as ρ (r ) = kr for r ≤ R =0 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 y introduction of the proton does not alter the negative charge distribution. x 1.28 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 (Fig 1.13). 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. d (b) Find the charges on β and γ after the collision. -Q q Q
- (c)Find the velocity of the plate γ after the collision and at a distance Fig. 1.13 d from the plate β. 1.29 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= rˆ r2 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 = × 10 –9 C [3] The number [3] actually arises from the speed of light in vaccum which is now taken to be exactly given by c = 2.99792458 × 108 m/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. Electric Charges and Fields (ii) Write 1 esu of charge = x C, where x is a dimensionless number. Show that this gives 1 10 –9 N.m 2 = 2 4π ∈0 x C2 With x = × 10 –9 , we have [3] 1 Nm 2 = [3]2 × 109 4π ∈0 C2 1 2 9 Nm or, = (2.99792458) × 10 (exactly). 4π ∈0 C2 1.30 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 x by x (x<<d) perpendicular to the line joining the two fixed charged as shown in Fig. 1.14. Show that q will perform simple harmonic – q d d –q oscillation of time period. Fig. 1.14 1/ 2 8π 3ε 0md 3 T = q2 1.31 Total charge –Q is uniformly spread along 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.