[2008EM/EVAL-Final]

$ \newcommand {\pieps}{4 \pi \epsilon_0} $

IMSc-IV Physics-IV : Electricity and Magnetism Jan-Apr 2008 MM : 60
End Semester Examination

       [$\oslash$] Question No. 1 is compulsory. Attempt FOUR more questions. All questions carry equal mark

      [$\oslash$]If extra questions are answered, the best required answers will be given credit.

      [$\oslash$] If any question is answered more than once, the first answer will be graded. 

1. Give mathemtical form for each of the following. 

• Coulomb's Law &
• Electrostatic forces are Conservative 
• Gauss Law &
• Biot Savart Law 
• No monopoles &
• Ampere's Law 
• Laws of Induction &
• Flux Rule 
• Charge Conservation &
• Displacement Current 
• Energy Density of Electric Field &
• Energy Density of Magnetic Field
• 1. Four charges $-q,q,-q,q$ are placed at $(-a,0),(-b,0),(b,0), (a,0)$ forming two electric dipoles whose effects do not cancel. Given expression for the electric potential on the line of the charges at $(r,0)$, and show for $(r\gg a > b)$ it has the form given by $$ \phi(r) =\frac{p}{\pieps r^2} +\frac{Q}{\pieps r^4} + \cdots $$ Hence find expressions for $p$ and $Q$ in terms of $q,a,b$.
  2. Find the electric field at the center of a semicircular arc which carries a charge density $+\sigma$ and $-\sigma$ as shown in figure.
• It is given that the electric potential due to thin spherical shell of radius $a$ and carrying charge $Q$ is $$ \phi(r) = \begin{cases} \frac{Q}{\pieps r} & r > a \\ \frac{Q}{\pieps a} & r < a \end{cases} $$ Using this result, or otherwise, compute the potential due to uniformly charged solid sphere having total charge $Q$ distributed uniformly over the volume of the sphere.
• 1. Show that the magnetic field due to a wire, of finite length and carrying a current $I$, at a point at distance $d$ has the magnitude $$ B=\frac{\mu_0}{4\pi d}(\sin\alpha_2-\sin\alpha_1) $$ % \qquad B_2=\frac{\mu_0}{2\pi d}(\cos\alpha_2-\cos\alpha_1)$$ where $\alpha_1,\alpha_2$ are angles as shown in figure below. Show that it reduces to the known result for infinite straight conductor.
  2. Compute the magnetic due to a rectangular loop, having sides $1$m, $\sqrt{3}$m, and carrying current $2$A at the center of the loop. \hfill[8+4]
• 1. An alpha particle travels in a circular path of radius $0.45$m in a magnetic field with $B=1.2$w/m$^2$. Calculate (a) its speed (b) its period of revolution, and (c) its kinetic energy.
  2. A circular coil is formed from a wire of length $L$ with $n$ turns. If the coil caries a current $I$ and is placed in an external magnetic field $B$. Show that maximum torque is developed when the number of turns is one and the value of the maximum torque is $\displaystyle \frac{L^2IB}{4\pi}$.
• 1. Describe the phenomenon of electromagnetic induction and Faraday's and Lenz's laws.
  2. For a conductor moving in a magnetic field, giving a simple example explain the origin of induced emf and derive the flux rule.
  3. How do you explain the appearance of induced emf when the conductor does not move but the magnetic field changes with time?
  4. Four circular disks, each having the same radius and thickness, are made to oscillate uniform magnetic field. If the initial amplitude is same in all cases rank the following four situations in terms of time taken for the oscillations to damp down completely, giving brief explanation for your answers.
  5. [(i)] copper disk with slits
  6. [(ii)] wooden disk
  7. [(iii)] aluminum disk without slits
  8. [(iv)] copper disk cooled in liquid nitrogen
• 1. What modification are needed in Maxwell's equations for time varying situations? How does the modified equation(s) explain the flux rule?
  2. Drive the the equation of continuity from the Maxwell's equations for the time varying case.
  3. Give examples of at least six results/concepts that require modifications in time varying situation.