problem 1:

a) A small sphere containing a positive charge of 2 C is hung vertically by an insulated thread. A second sphere having a negative charge of – 2 C is placed at a distance d from the original position of the first sphere (when it hangs vertically) on the horizontal line joining the two spheres. Where a third sphere carrying a positive charge of 4 C should be placed on the horizontal line so that the first sphere remains hanging vertically?

b) Consider a long well conducting cylindrical pipe of length L. Its inner radius is r₁ and outer radius is r₂ in such a way L is much larger than r₂. Assume that an amount of positive charge q is released on its inner surface by briefly touching the inner wall with a charged body. Find out the electric field in the regions

i) r < r₁,
ii) r₁ < r < r₂ and
iii) r > r₂.

c) Two charges of +100 µC are placed on diagonally opposite corners of a square with sides of length 1.0 m long. A charge of − 50 µC is placed at a third corner.

i) Find out the potential energy of this configuration supposing that all three charges were initially at infinity.
ii) Compute the potential at the fourth corner of the square, supposing that U = 0 at infinity.

problem 2:

a) Describe the phenomenon of polarization of a dielectric. Show that, when a dielectric material is filled between the plates of a capacitor, the value of capacitance increases by factor of ε_r, the relative permittivity of the dielectric.

b) The capacitance of a parallel plate capacitor is raised by a factor of 8 when a dielectric material fills the space between its plates. What is the relative permittivity of the dielectric material? If this material is placed in between the plates of a cylindrical capacitor of outer and inner radii 14 cm and 12 cm, correspondingly, compute the capacitance per unit length of the cylindrical capacitor.

c) A glass of relative permittivity 4 is kept in an external electric field of magnitude 10²Vm⁻¹. Compute the polarization vector, atomic polarisability and the refractive index of the glass.

d) The energy of a capacitor is 3.0 µJ after it has been charged by a 1.5 V battery. Compute its energy when it is charged by a 3.0 V battery.

problem 3:

a) Distinguish between thermal velocity and drift velocity of electrons in a conductor placed in an electric field. A copper wire of diameter 2 mm and length 50 m is connected across a battery of 2 V. Compute the current density in the wire and drift velocity of the electrons. The resistivity of copper is 1.72 × 10⁻⁸ ?m and n = 8.0 × 10²⁸ electrons m⁻³.

b) A horizontal, straight wire carrying 20 A current from west to east is in the earth’s magnetic field B. At this place, B is parallel to the surface of the earth, points to the north and its magnitude is 0.05 mT. Find out the magnetic force on 1 m length of the wire. If mass of this length of wire is 50 g, compute the value of current in the wire so that its weight is balanced by the magnetic force.

c) An electron, travelling horizontally with a speed of 2.5 × 10⁶ ms⁻¹, enters a region of uniform electric field of magnitude 500 NC −1 directed upward. The field extends horizontally for a distance of 60 cm. Compute:

i) The vertical displacement of the electron, and
ii) The velocity of the electron as it emerges from the region of the field.

d) Why do magnetic moments, occurring due to spins of ferromagnetic materials, tend to align parallel to one other? Establish the relation B = µ₀ (H + M) for a ferromagnetic material.

problem 4:

a) By using Maxwell’s equations in free space, derive the wave equation for the z-component of the electric field vector.

b) A uniform plane wave of 100 MHz travelling in free space strikes a big block of a material having ε = 4 ε₀, µ = 9 µ₀ and σ = 0 normal to the surface. If the incident electric field vector is given by:

E = 3000 cos (ωt – βy) zˆ Vm⁻¹

prepare down the complete expressions for the incident, reflected and transmitted field vectors.