CHAPTER - 1 ELECTRIC CHARGES AND FIELDS

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.

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
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.
(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.

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?

A paisa coin is made up of Al-Mg alloy and weighs 0.75g. It has a
square shape and its diagonal measures 17 mm. It is electrically
neutral and contains equal amounts of positive and negative
charges.
Treating the paisa coins made up of only Al, find the magnitude of
equal number of positive and negative charges. What conclusion
do you draw from this magnitude?

If the total charge enclosed by a surface is zero, does it imply that
the elecric field everywhere on the surface is zero? Conversely, if
the electric field everywhere on a surface is zero, does it imply that
net charge inside is zero.

The dimensions of an atom are of the order of an Angstrom. Thus
there must be large electric fields between the protons and electrons.
Why, then is the electrostatic field inside a conductor zero?

The dimensions of an atom are of the order of an Angstrom. Thus
there must be large electric fields between the protons and electrons.
Why, then is the electrostatic field inside a conductor zero?

A metallic spherical shell has an inner radius R1 and outer radius
R2. A charge Q is placed at the centre of the spherical cavity. What
will be surface charge density on (i) the inner surface, and (ii) the
outer surface?

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

Two identical parallel plate (air) capacitors C1 and C2 have capacitances C each. The between their plates is now filled with dielectrics as shown. If the two capacitors still have equal capacitance, obtain the relation between dielectric constants K, K1 and K2 .

Draw 3 equipotential surfaces corresponding to a field that uniformly increases in magnitude but remains constant along Z-direction. How are these surfaces different from that of a constant electric field along Z-direction?

Show that the electric field at the surface of a charged conductor is given by E=/ε₀ n, where is the surface charge density and  n is a unit vector normal to the surface in the outward direction.

. Name the physical quantity whose S.I. unit is JC1. Is it a scalar or a vector quantity?

Net capacitance of three identical capacitors in series is 1 F. What will be their net capacitance if connected in parallel? Find the ratio of energy stored in the two configurations if they are both connected to the same source.

Plot a graph showing the variation of coulomb force (F) versus 1/r² , where r is the distance between the two charges of each pair of charges: (1 C, 2 C) and (2 C – 3 C). Interpret the graphs obtained.

The dipole is aligned parallel to the field. Find the work done in rotating it through the angle

of 180°

A charge ‘q’ is placed at the centre of a cube of side l. What is the electric flux passing through

each face of the cube?

Draw the equipotential surfaces due to an electric dipole. Locate the points where the potential

due to the dipole is zero.

A slab of material of dielectric constant K has the same area as that of the plates of a parallel plate

capacitor but has the thickness d/2, where d is the separation between the plates. Find out the

expression for its capacitance when the slab is inserted between the plates of the capacitor.

Two charges of magnitudes – 2Q and + Q are located at points (a, 0) and (4a, 0) respectively. What is the electric flux due to these charges through a sphere of radius ‘3a’ with its centre at the origin?

A parallel plate capacitor of capacitance C is charged to a potential V. It is then connected to another uncharged capacitor having the same capacitance. Find out the ratio of the energy stored in the combined system to that stored initially in the single capacitor.

Why do the electric field lines never cross each other ?

Suppose that the particle in Exercise in 1.33 is an electron projected with velocity vx= 2.0 x 106 m s-1. If Ebetween the plates separated by 0.5 cm is 9.1 x 102 N/C, where will the electron strike the upper plate? (| e | =1.6 x 10-19 C, me = 9.1 x 10-31 kg.)

A particle of mass m and charge ( - q) enters the region between the two charged plates initially moving along x-axis with speed vx (like particle 1 in Fig. 1.33). The length of plate is L and an uniform electric field E is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is qEL2/ (2m).

Compare this motion with motion of a projectile in gravitational field discussed in Section 4.10 of Class XI Textbook of Physics.

(a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., where E = 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable.

(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.

It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks. A proton and a neutron consist of three quarks each. Two types of quarks, the so called 'up' quark (denoted by u) of charge (+2/3) e, and the 'down' quark (denoted by d) of charge (-1/3) e, together with electrons build up ordinary matter. (Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron.

Obtain the formula for the electric field due to a long thin wire of uniform linear charge density ÃŽÂ»without using Gauss's law. [Hint: Use Coulomb's law directly and evaluate the necessary integral.]

A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is , where is the unit vector in the outward normal direction, and  is the surface charge density near the hole.

(a) A conductor A with a cavity as shown in Fig. 1.36(a) is given a charge Q. Show that the entire charge must appear on the outer surface of the conductor. (b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A is Q + q [Fig. 1.36(b)]. (c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.

In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of 105 NC-1 per metre. What are the force and torque experienced by a system having a total dipole moment equal to 10-7 Cm in the negative z-direction?

Which among the curves shown in Fig. 1.35 cannot possibly represent electrostatic field lines?

(a)

(b)

(c)

(d)

(e)

An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 x 104 N C-1 in Millikan's oil drop experiment. The density of the oil is 1.26 g cm-3. Estimate the radius of the drop. (g = 9.81 m s-2;e = 1.60 x 10-19 C).

Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 x 10-22 C/m2. What is E: (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates?

An infinite line charge produces a field of 9 x 104 N/C at a distance of 2 cm. Calculate the linear charge density.

A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 μC/m2. (a) Find the charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere?

A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 x 103 N/C and points radially inward, what is the net charge on the sphere?

A point charge causes an electric flux of -1.0 x 103 Nm2/C to pass through a spherical Gaussian surface of 10.0 cm radius centered on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge?

A point charge of 2.0 μC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface?

A point charge +10 μC is a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge 10 cm.)

Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 x 103 N m2/C. (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not?

What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes?

Consider a uniform electric field E = 3 x 103 îN/C. (a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?

Figure 1.33 shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?

Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?

(a) Two insulated charged copper spheres A and B have their centers separated by a distance of 50 cm. What is the mutual force of electrostatic repulsion if the charge on each is 6.5 x 10-7 C? The radii of A and B are negligible compared to the distance of separation.

(b) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved?

A polythene piece rubbed with wool is found to have a negative charge of 3 x 10-7 C.

(a) Estimate the number of electrons transferred (from which to which?)

(b) Is there a transfer of mass from wool to polythene?

An electric dipole with dipole moment 4 x 10-9 C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 x 104 N C-1. Calculate the magnitude of the torque acting on the dipole.

A system has two charges qA = 2.5 x 10-7 C and qB = -2.5 x 10-7 C located at points A: (0, 0, - 15 cm) and B: (0, 0, + 15 cm), respectively. What are the total charge and electric dipole moment of the system?

Two point charges qA = 3 μC and qB = -3 μC are located 20 cm apart in vacuum.

(a) What is the electric field at the midpoint O of the line AB joining the two charges?

(b) If a negative test charge of magnitude 1.5 x 10-9 C is placed at this point, what is the force experienced by the test charge?

(a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?

(b) Explain why two field lines never cross each other at any point?

Four point charges qA = 2 μC, qB = -5 μC, qC = 2 μC, and qD = -5 μC are located at the corners of a square ABCD of side 10 cm. What is the force on a charge of 1 μC placed at the centre of the square?

When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge.

(a) Explain the meaning of the statement 'electric charge of a body is quantised'.

(b) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges?

Check that the ratioke2/G mempis dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

The electrostatic force on a small sphere of charge 0.4 μC due to another small sphere of charge - 0.8 μC in air is 0.2 N. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first?

What is the force between two small charged spheres having charges of 2 x 10-7 C and 3 x 10-7 C placed 30 cm apart in air?