Chapter 1: Electric Charges and Fields

Consider a coin of example 1.20. It is electrically neutral and contains equal amounts of positive and negative charge of magnitude 34.8 kC. Suppose that these equal charges were concentrated in two-point charges separated by

a) 1 cm

b) 100 m

Find the force on each such point charge in each of the three cases. What do you conclude from these results?

Figure represents a crystal unit of caesium chloride, CsCl. The caesium atoms, represented by open circles are situated at the corners of a cube of side 0.40 nm, whereas a Cl atom is situated at the centre of the cube. The Cs atoms are deficient in one electron while the Cl atom carries an excess electron

a) what is the net electric field on the Cl atom due to eight Cs atoms?

b) suppose that the Cs atom at the corner A is missing. What is the net force now on the Cl atom due to seven remaining Cs atoms?

Two charges q and -3q are placed fixed on x-axis separated by distance ‘d’. Where should a third charge 2q be placed such that it will not experience any force?

Figure shows the electric field lines around three-point charges A, B, and C

a) which charges are positive?

b) which charge has the largest magnitude? Why?

c) in which region or regions of the picture could the electric field be zero?

i) near A

ii) near B

iii) near C

iv) nowhere

Five charges, q each are placed at the corners of a regular pentagon of side ‘a’

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 corners is removed?

iii) what will be the electric field at O if the charge q at A is replaced by –q?

b) how would you answer to a) be affected if pentagon is replaced by an n-sided regular polygon with charge q at each of its corners?

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.

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

Consider a sphere of radius R with charge density distributed as

ρ(r) = kr for r ≤ R

ρ(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 introduction of the proton does not alter the negative charge distribution

Two fixed, identical conducting plates (α and 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 redistributed charge among β and γ.

a) find the electric field acting on the plate γ before collision

b) find the charges on β and γ after the collision

c) find the velocity of the plate γ after the collision and at a distance d from the plate β

There is another useful system of units, besides the SI/mks A system called the cgs system. In this system Columb’s law is given by

Where the distance r is measured in cm, F in dynes and the charges in electrostatic units where 1 es unit of charge = 1/[3] × 10^-9 C. The number [3] actually arise from the speed of light in vacuum 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.

a) show that the Columb’s law is 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

b) Write 1 csu of charge = xC where x is a dimensionless number. Show that this gives

1/4πε₀ = 10^-9/x² N.m²/C²

With x = 1/[3] × 10⁹ Nm²/C²

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 perpendicular to the line joining the two fixed charged as shown in the figure. Show that q will perform simple harmonic oscillation of time period.