ASSIGNMENT -2 SUBJECT: ENGINEERING ELECTROMAGNETICS(15EC36) TOPIC: GAUSS'S LAW AND DIVERGENCE FACULTY: BENAK PATEL M P 1. State and prove and Gauss's law. 2. Show that ρs outer = (-a/b) ρs inner. 3. Calculate D in rectangular coordinates at point P(2,-3,6) produced by a) a point charge Q= 55mc at Q(-2,3,-6). b) a uniform line charge ρL=20mc/m on the x-axis. c) a uniform surface charge density ρs=120μC /m2 on the plane z= -5m. 4. Given the electric flux density, D= 0.3r2ar nc/m2 in free space. a) Find E at point P(2,250,900) b) Find the total charge within the sphere r=3. c) Find the total electric flux leaving the sphere r=4. 5. In each of the following parts, find a numerical value for div D at the point specified a) D= (2xyz-y2)ax + (x2z-2xy)ay+x2yaz c/m2 at P(2,3,-1). b) D=2rz2sin2Φ ar + rz2 sin2ΦaΦ +2r2zsin2Φaz c/m2 at P(2,1100,-1). c) D=2r sinθ cosΦ ar + r cosθ cosΦ aθ - rsinΦaΦ c/m2 at P(1,300,500). 6. State Gauss's law in point form. 7. Determine an expression for the volume charge density associated with each D field following a) D= 5sinθ aθ + 5sinΦ aΦ c/m2 at P(0.5,π/4,π/4). b) D= (4xy/z) ax + (2x2/z) ay +- (2x2y/z2) az. c) D= z sinΦar + cosθ sinΦ aθ + rsinΦaz . 8. State and prove divergence theorem. 9. Given D=5r ar c/m2, prove divergence theorem for a shell region enclosed by spherical surface at r=a and r=b (b>a) and centered at the origin. 10. Given D=30e-rar - 2zaz c/m2 in cylindrical coordinates. Evaluate both sides of the divergence theorem for volume enclosed by r=2, z=0, z=5.

TOPIC: ENERGY,POTENTIAL AND CONDUCTOR 1. Find the work done in moving a charge of 2 C from (2,0,0)m to (0,2,0) m along the straight line path joining two points, if the electric field is E= 12xax-4yay v/m. 2. Given the electric field E= (6xyzax + 2x2zay-4x2yaz)v/m , find the differential amount of work done in moving a 3nc charge a distance of 5μm,starting at P(1,-2,3) and proceeding in the direction aL a) -6/7 ax+2/7 ay+5/7az b) 6/7 ax- 4/7ay-5/7 az 3. Find V, E and D in free space a) V= Eoe-xsin( πy/4) in cartesian at P(0,0,1) . b) V= 100rcosΦ /(z2+1) at Q(4,500,3). c) V= 60sinθ/r2 at R(3,600,250) 4. Derive the potential due to point charge. 5. Derive the potential due to several point charges. 6. A point charge of 6nc is located at origin in free space, find potential of point P if P is located at (0.2,-0.4,0.4) and a) V=0 at infinity b) V=0 at (1,0,0) c) V=20V at (-0.5,1,-1). 7. Derive an expression for J, ρv and velocity of the charge element. 8. With usual notations derive the expressions ᴧ.J= -∂ ρv /∂ t. 9. Find the current a) J= 2x2ax 2xy3ay + 2xyaz A/m2 in outward direction of a cube of 1m, with one corner at the origin and edges parallel to the coordinate axes. b) J= 10e-100 ra A/m2 0.01≤ r ≤0.02 ,0≤ z ≤1m. Φ

c) J= 100cosθar/ r2+1 A/m2 r=3m, 0<θ < π/6, 0< Φ< 2π.