Electricity and Magnetism — DU B.Sc. (Hons) Physics Previous Year Question Papers

4. Answer any four of the remaining five questions. (a) Two negative charges of unit magnitude and a positive charge 𝑞𝑞 are placed along a straight line. At what position and for what value of 𝑞𝑞 will the system be in equilibrium? Discuss if it is stable or unstable equilibrium. (b) The linear charge density on a rod of length 𝐿𝐿 is given by 𝜆𝜆 (𝑥𝑥) = 𝜆𝜆0 + 𝑥𝑥 is a constant and 𝑥𝑥 is the distance from one end of the rod. Find the total charge on the rod. (c) A charge −𝑞𝑞 is placed in vacuum at the point (𝑑𝑑, 0,0), where 𝑑𝑑 > 0. The region 𝑥𝑥 ≤ 0 is filled uniformly with a conductor. Find the electric field at the point �𝒅𝒅 (d) Given that 𝐸𝐸 (𝑉𝑉/𝑚𝑚) at the charge-free dielectric interface between two different dielectric materials with relative permittivity’s 𝜖𝜖1 = 4 and 𝜖𝜖2 = 3 respectively. Find the electric field 𝐸𝐸 2 and electric displacement 𝐷𝐷 2 in the second dielectric. (e) A magnetic field is given by the expression 𝐵𝐵 , where a, b and c are some constants. Find a relation between 𝑎𝑎 and 𝑏𝑏. (f) A vector potential in a region is given by 𝐴𝐴 = 𝑎𝑎𝑎𝑎2𝚤𝚤̂. Find the volume current density in the region. (g) Write down Maxwell’s equations for free space in differential form explaining their physical significance. (a) The electric potential of a charge configuration is expressed as 𝑉𝑉(𝑟𝑟) = 𝐴𝐴 𝑒𝑒−𝜆𝜆𝜆𝜆 𝑟𝑟 , where 𝐴𝐴 and 𝜆𝜆 are constants. Find the electric field 𝐸𝐸 � ⃗(𝑟𝑟), the charge density 𝜌𝜌(𝑟𝑟), and total charge (b) State and prove First uniqueness theorem. (7) (c) Use Laplace’s equation to find the potential for the region between two concentric spheres when 𝑉𝑉 1 = 0 volt at 𝑟𝑟 = 2 𝑚𝑚 and 𝑉𝑉 2 = 150 volt at 𝑟𝑟 = 3 𝑚𝑚. (3) 3. (a) A conducting sphere of radius 𝑅𝑅, is placed at the origin. The point charges ±𝑞𝑞 are located at distance ∓𝑑𝑑 from the centre of the sphere (with 𝑑𝑑 > 𝑅𝑅). Using method of images, determine the magnitude and positions of the image charges. Also, find the electric potential at an arbitrary field point 𝑃𝑃 due to the real charges and image charges. (8) (c) Thin dielectric rod of cross-section 𝐴𝐴 extends along 𝑎𝑎 −axis from 𝑎𝑎 = 0 to 𝑎𝑎 = 𝐿𝐿 . The polarization of the rod is along its length and is given by 𝑃𝑃 , where 𝑎𝑎 and 𝑏𝑏 are some constants. Obtain the bound volume and surface charge densities at each end of the rod. (3) 4. (a) Derive an expression for magnetic field along the axis of two identical coaxial coils of radius 𝑅𝑅 and turns 𝑁𝑁, separated by a distance 𝑑𝑑 and carrying current 𝐼𝐼 in the same direction. Plot a graph showing the variation of the magnetic field along the axis. (b) Find the magnetic field of an infinite uniform surface current described by surface current density 𝐾𝐾 = 𝐾𝐾𝚤𝚤̂ flowing over 𝑎𝑎 = 0 plane. 6. (c) Explain the significance of continuity equation in electromagnetism. (3) 5. (a) Starting from Biot-Savart’s law for a volume current, show that