Electric potential at an equatorial point of a small dipole with dipole moment $P$ ($r$,distance from the dipole) is

$A$ closed pipe is suddenly opened and changed to an open pipe of the same length. The fundamental frequency of the resulting open pipe is less than that of the $3^{rd}$ harmonic of the earlier closed pipe by $55 ~Hz$. Then,the value of the fundamental frequency of the closed pipe is: (in $~Hz$)

$A$ long solenoid has $200$ turns per cm and carries a current $i$. The magnetic field at its centre is $6.28 \times 10^{-2} ~Wb/m^2$. Another long solenoid has $100$ turns per cm and it carries a current $\frac{i}{3}$. The value of the magnetic field at its centre is nearly:

If the impurity in a metal has a greater affinity for oxygen and is more easily oxidised than the metal,then the purification of metal may be carried out by

$A$ total charge $Q$ is broken into two parts $Q_1$ and $Q_2$ and they are placed at a distance $R$ from each other. The maximum force of repulsion between them will occur when:

If the coefficient of $x$ in the expansion of $\left(x^2+\frac{k}{x}\right)^5$ is $270$,then $k$ is equal to