Course Code: BPHCT-133

Assignment Code: BPHCT-133/TMA/2023

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

\section{PART A}

1. a) Obtain the directional derivative for a scalar field \(\phi(x, y, z)=5 x y-3 y^{2} z^{3}\) at the point \((0,1,-1)\) in the direction \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\).

b) Calculate the work done by a force \(\overrightarrow{\mathrm{F}}=4 y^{3} z^{3} \hat{\mathbf{i}}-2 x^{2} y z^{3} \hat{\mathbf{j}}+3 x y^{2} z^{3} \hat{\mathrm{k}}\) in moving a particle along the path defined by the equations \(x=t ; y=t ; z=t\) between the points \(t=0\) and \(t=1\).

c) Using the divergence theorem evaluate \(\iint \overrightarrow{\mathbf{A}} \cdot \hat{\mathbf{n}} d S\), where

\(\overrightarrow{\mathbf{A}}=2 x \cos ^{2} y \hat{\mathbf{i}}+3 x z \hat{\mathbf{j}}+2 z \sin ^{2} y \hat{\mathbf{k}}\), over the surface of a sphere with its centre at the origin and a radius of 8 units.

2. a) Determine the electrostatic force and electrostatic field on a charged particle located at \(A\) in the Figure given below due to the charged particles situated at \(B\) and \(C\). The value of the charge on each of these particles is indicated in the Figure below:


Express your result both in the unit vector notation and as magnitude.

b) Explain with the help of diagrams what spherically and cylindrically symmetric charge distributions are. What is the electric field at a point inside a hollow metallic sphere of radius \(R\) having volume charge density \(\rho\) ?

c) Two particles carrying \(4 \mathrm{C}\) and \(-2 \mathrm{C}\) charges are placed on a \(1 \mathrm{~m}\) long straight wire. Determine the point on the line joining these particles where the electric potential is zero.

\section{PART B}

3. a) Explain the phenomenon of polarisation 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 \(K\), the dielectric constant of the material.

b) Determine the value of equivalent capacitance between points \(A\) and \(B\) for the combination of capacitors shown in the Figure below:


c) The energy of a capacitor is \(4.0 \mu \mathrm{J}\) after it has been charged by a \(1.5 \mathrm{~V}\) battery. Calculate its energy when it is charged by a \(6.0 \mathrm{~V}\) battery.

4. a) Obtain an expression to show that the change in the quantity of charge enclosed in an arbitrary volume is accompanied by a net flow of charge inwards or outward across the surface of the enclosed volume.

b) A horizontal, straight wire carrying 12.0 A current from west to east is in the earth’s magnetic field \(\mathbf{B}\). At this place, \(\mathbf{B}\) is parallel to the surface of the earth, points to the north and its magnitude is \(0.04 \mathrm{mT}\). Determine the magnetic force on \(1 \mathrm{~m}\) length of the wire. If mass of this length of wire is \(50 \mathrm{~g}\), calculate the value of current in the wire so that its weight is balanced by the magnetic force.

c) Using Maxwell’s equations in free space, derive the wave equation for the magnetic field vector.