## Field Theory - Dec 2014

### Electronics & Communication (Semester 3)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1(a)** State vector form of Coulomb's law of force between two point charges and indicate the units of the quantities in the equation.(6 marks)
**1(b)** State and prove Gauss law for point charge.(6 marks)
**1(c)** Two point charges,Q_{1} and Q_{2} are located at (1,2,0)_{m} and (2,0,0)_{m} respectively.Find the relation between the charges Q_{1} and Q_{2} such that the total force on a unit positive charge at (-1,1,0) have i) no x-component.(8 marks)
**2(a)** Define potential difference and absolute potential.(4 marks)
**2(b)** Establish the relation $$E=\bigtriangleup \vee.$$(6 marks)
**2(c)** Electrical potential at an arbitrary point free - space is given as :

$$V-(x+1)^{2}+(y+2)^{2}+(z-3)^{2}. At \ P (2,1,0)\ find\\ i) v\ ii) E^{\rightarrow }\ iii) E^{\rightarrow }\ iv) D^{\rightarrow }\ v) D^{\rightarrow }\ vi) P_{v}.$$(10 marks)
**3(a)** Derive the expression for Poisson's equation.(4 marks)
**3(b)** Write the expression for Laplace's equation in cylindrical and spherical coordinates.(4 marks)
**3(c)** State and prove uniqueness theorem.(6 marks)
**3(d)** Given the potential field$$V=x^{2}yz-ky^{3}z$$ volts :

i) Find k if potential field satisfies Laplace's equation

ii) find$$E^{\rightarrow }$$at (1,2,3).(6 marks)
**4(a)** Starting form Biot-Savort's law, derive the expression for the magnetic field intensity at a point due to finite length current carrying conductor.(8 marks)
**4(b)** Verify stoke's theorem for the field $$\underset{H}{\rightarrow}$$_$$2rcos\Theta a\ r^{\vee }+ra\Theta ^{\wedge }$$ for the path shown r=0 to 1; 0 to $$90^{0}$$
(8 marks)
**4(c)** Explain scalar and vector magnetic potenial.(4 marks)
**5(a)** Derive expression for magnetic force on :

i) Moving point charge

Differential current element.(10 marks)
**5(b)** A current element$$I_{1}dI_{2}-10^{-4}\ \widehat{a_z}$$ (AM) is located at$$P_{1}$$(-2,0,0).Both are in free space :

Find force exerted on $$I_{2}d1_{2}-10^{-6}[{\widehat{ax}}-2\widehat{ay}+3\widehat{az}](Am)$$ is located at $$P_{2}$$(-2,0,0). Both are in free space:

i) Find force exerted on $$I_{2}dl_{2}by I_{2}dI_{1}$$

ii) Find force exerted on$$I_{1}dl_{1}by I_{2}dI_{2}.$$(10 marks)
**6(a)** List Maxwell's equations in point form and lntergral form.(8 marks)
**6(b)** A homogeneous material has ?=2×1$$\epsilon =2×10^{6} F/M and\ \mu =1.25×10^{5}$$ and$$\sigma =0.$$.Electric field intensity $$\overrightarrow{E}$$=400 cos$$(10^{9}t-kz)a\widehat{x}\ V/m$$. If all the field vary sinsoidally,find$$\overrightarrow{D},\overrightarrow{B},\overrightarrow{H} and k using Maxwell's equations.\lt/span\gt\ltspan class='paper-ques-marks'\gt(12 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(b)\lt/b\gt State and explain Poynting theorem.\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7(c)\lt/b\gt Starting form Maxwell's equations derive wave equation in E and H for a uniform plane wave travelling in free space.\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8(a)\lt/b\gt Write short notes on :\ltbr\gt i) SWR and reflection coefficient \ltbr\gt ii) Skin depth.\lt/span\gt\ltspan class='paper-ques-marks'\gt(10 marks)\lt/span\gt
\lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt8(b)\lt/b\gt A Ghz plane wave in free space has electric field intensity 15 V/m. Find: \ltbr\gt i) Velocity of propagation \ltbr\gt ii) Wavelength\ltbr\gt iii) Characteristic impedance of the medium \ltbr\gt iv) Amplitude of magnetic field intensity \ltbr\gt v) Propagation constant$$\beta $$.(10 marks)