Question

Question 1. (a) Write down the differential form of Maxwell's equations in matter for the dynamic case (where the electric and magnetic field can change with time), in the presence of free charges and currents. Describe all physical quantities and constants used. [10] (6) (b) Write down the integral form of Ampere's law in vacuum for the static (non time- dependent) case. Using Stokes' theorem, derive the differential form of Ampere's law. [4] (c) Two charges 91= 5 uC and 42=-5 uC are located at Fi = (1,0,0) m and r2 = (0,0,-1) m. What is the force on qı? Write it in a vector form. What is the electric dipole moment of this configuration? Write it in a vector form. What are the electric fluxes through two spherical surfaces centered on the origin with radii R = 0.5 m and R2= 10 m? What is the potential energy of this charge configuration? (d) The magnitude 111 of the current density in a certain wire with a circular cross section of radius R = 2.00 mm is given by lil = (3.00 x 108) r2, with lil in Amperes per square meter and the radial distance r in meters. What is the current through the outer section bounded by r = 0.9R and r = R? (e) Show that the force of attraction between the plates of a parallel plate capacitor is given by F = E,AV2/(2d2), where A is the area, d the distance of separation, V the potential difference between the positively and negatively charged plates and Eg the permittivity. [5] (1) Earth has a radius of 6.37 x 10ºm and a magnetic dipole moment of 8.0 x 1022 J/T. What current would have to be produced in a single turn of a wire extending around Earth at its geomagnetic equator if we wished to setup such a dipole? Could such an arrangement be used to cancel out Earth's magnetism i) at points in space well above Earth's surface or ii) on Earth's surface? (g) An electrostatic potential is given by V = V. (x2 + y2 + z2)xyz. Derive the corresponding electric field. Derive the charge distribution which generates this field.

Answer 1

a) Maxwell's eguations in matter are )E-- iv) V x +eu Whert E= elechic field charge clansity E = permittivity of ses matter magmetized magnetic field ilpermea bility of matteY J= CHTTT dansi t DElectric fldis a charge where the force would be exerted on other charged particles gon oeround a charge density () i Charge enclosed in a volume unit

e is permi tivity of medtun eleetric Plux Passin per unitare relatve pemith viy E Ex E Where, E ermitvity ot fvacuum is the magne tized magnetie field, whch means the / magnetic field pasing thoough the matte such that where Bis appliedl magnetic feld pe meability f matter Cmedium) i-e magnetie Flux pasi ng ORYunit area of maey. whare y aive permeabilihy Mpermeability ofvacuun 1257 x106 NA

Cument dunsihy J is change Plowing per unit volume per unit atime b) In vacuum, B=H So, J=o And Ampere's law in integral form is where is elechic Plux using Stokes ttheorem, 3di S(B). d3 d3 area vechor .. Ampere's law beiomes, B taking deivatives of both sides (7x3)= V(464) t t E v4= E) iLu