Square Loop in a Magnetic Field

Problem:

A rectangular frame ABCD made of auniform metal wire has a straight connection between E & F made of the same wire as shown in the figure. AEFD is a square of side 1 m & EB = FC = 0.5 m. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the place ofthe paper & normal to it. The rate of change of the magnetic field is 1 T/s, the resistance per unit length ofthe wire is 1 $\Omega$/m. Find the current in segments AE, BE & EF. [IIT JEE'93]

Electron entering a capacitor

Problem:

An electron enters the region between the plates of a parallel plate capacitor at a point equidistant from either plate. The capacitor plates are $2* 10^{-2}$ m apart & $10^{-1}$ m long. A potential difference of 300 volt is kept across the plates. Assuming that the initial velocity of the electron is parallel to the capacitor plates, calculate the largest value of the velocity of the electron so that they do not fly out of the capacitor at the other end. [IIT JEE]

Capacitance

Problem:

The capacitance of a parallel plate capacitor with plate area $A$ & separation $d$ is $C$. The space between the plates is filled with two wedges of di-electric constant $K_{1}$ & $K_{2}$ respectively. Find the capacitance of the resulting capacitor. [IIT JEE'96]

Mechanics question from Book by S S Krotov

Problem:

Small balls with zero initial velocity fall from a height $H = \displaystyle\frac{R}{8}$ near the vertical axis of symmetry on a concave spherical surface of radius $R$. Assuming that the impacts of the balls against the surface are perfectly elastic, prove that after the first impact each ball gets into the lowest point of the spherical surface.

Conductor Harmonic Motion - IIT JEE 2003

Problem:

Along horizontal wire AB, which is free to move in a vertical plane and carries a steady current of 20 A, is in equilibrium at a height of 0.01 m over another parallel long wire CD which is fixed in a horizontal plane and carries a steady current of 30 A. Show that when AB is slightly depressed, it executes simple harmonic motion. Find the period of oscillations.

Find value of Resistance - IIT JEE 2005

Problem:

In the figure both cells A and B are of equal emf. Find R for which potential difference across battery A will be zero, long time after the switch is closed. Internal resistance of batteries A and B are $r_{1}$ and $r_{2 }$respectively ($r_{1}>r_{2 }$).

Kinetic Theory of Gases - IIT JEE 1993

The following problem appeared in the Physics paper of IIT JEE 1993. It still considered to be one of the most formidable problems ever asked in the history of the exam.

Problem:

A neutron of kinetic energy 65 eV collides in-elastically with a singly ionized helium atom at rest. It is scattered at an angle of 90º with respect of its original direction.

(i) Find the allowed values of the energy of the neutron and that of the atom after the collision.

(ii) If the atom get de-excited subsequently by emitting radiation, find the frequencies of the emitted radiation. [Given: mass of He atom =4×(mass of neutron),Ionization energy of He atom = 13.6 eV]

Mutual Inductance

One book every physics or engineering student must have in his collection is Aptitude Problems in Physics which is a collection of Moscow Physics Olympiad problems edited by S.S Krotov. The problems in the book are absolute gems. I will post a few questions from the Electricity and Magnetism section.

Problem:

Two long cylindrical coils with uniform winding of the same length and nearly the same radius have inductance $L_{1}$ and $L_{2}$. The coils are co-axially inserted into each other and connected to a current source. The directions of the current in the coils is such that the fluxes add . Determine the inductance L of such a composite coil.

Solution:

From the definition of inductance we have $\displaystyle L=\frac{N\phi}{I}$

$\displaystyle\phi =\frac{MMF}{Reluctance}=\displaystyle\frac{NIA}{\mu_{o}l}\implies L_{1}=kN_{1}^2$  and $L_{2}=kN_{2}^2$

When the coils are coaxially combined with flux additive composite inductance

$L_{c}=k(N_{1}+N_{2})^2$

$L_{c}=\displaystyle k\left(\sqrt{\frac{L_{1}}{k}}+\sqrt{\frac{L_{2}}{k}}\right)^2=L_{1}+L_{2}+2\sqrt{L_{1}L_{2}}$