Magnetic Field Questions

An electron (mass $9\times 10^{-31}\text{kg}$ and charge $1.6\times 10^{-19}\text{C}$) moving with speed $c/100$ ($c$ = speed of light) is injected into a magnetic field $\vec{B}$ of magnitude $9\times 10^{-4}\text{T}$ perpendicular to its direction of motion. We wish to apply a uniform electric field $\vec{E}$ together with the magnetic field so that the electron does not deflect from its path. Then (speed of light $c=3\times 10^8{\text{ms}}^{-1}$):

The electric field in a plane electromagnetic wave is given by $E_z=60\cos (5x+1.5\times 10^{9}t)\text{ V/m}$. Then expression for the corresponding magnetic field is:

Under the influence of a uniform magnetic field a charged particle is moving in a circle of radius $R$ with constant speed $v$. The time period of the motion:

A particle of mass $m$, charge $Q$, and kinetic energy $T$ enters a transverse uniform magnetic field of induction $\vec{B}$. After $3$ seconds, the kinetic energy of the particle will be:

Under the influence of a uniform magnetic field, a charged particle moves with constant speed $V$ in a circle of radius $R$. The time period of rotation of the particle:

A thin ring of radius $R\text{ meter}$ has charge $q\text{ coulomb}$ uniformly spread on it. The ring rotates about its axis with a constant frequency of $f\text{ revolutions/s}$. The value of magnetic induction in ${\text{Wb/m}}^2$ at the centre of the ring is:

Two identical long conducting wires $AOB$ and $COD$ are placed at right angle to each other, with one above other such that '$O$ ' is their common point. The wires carry $I_1$ and $I_2$ currents, respectively. Point '$P$ ' is lying at distance '$d$ ' from '$O$ ' along a direction perpendicular to the plane containing the wires. The magnetic field at point '$P$ ' will be: