$A$ long straight wire,of radius $a$,carries a current distributed uniformly over its cross-section. The ratio of the magnetic fields due to the wire,at distance $\frac{a}{3}$ and $2a$ respectively from the axis of the wire,is

At a given instant of time,two particles have position vectors $4 \hat{i} + 4 \hat{j} + 57 \hat{k} \text{ m}$ and $2 \hat{i} + 2 \hat{j} + 5 \hat{k} \text{ m}$ respectively. If the velocity of the first particle is $0.4 \hat{i} \text{ ms}^{-1}$,what is the velocity of the second particle in $\text{ms}^{-1}$ if they collide after $10 \text{ s}$?

$A$ particle of mass $m_1$ is moving with a velocity $v_1$ and another particle of mass $m_2$ is moving with a velocity $v_2$. Both of them have the same momentum,but their kinetic energies are $E_1$ and $E_2$ respectively. If $m_1 > m_2$,then:

$A$ galvanometer of resistance $50 \Omega$ is connected to a battery of $3 \text{ V}$ along with a resistance of $2950 \Omega$ in series. $A$ full-scale deflection of $30$ divisions is obtained in the galvanometer. In order to reduce this deflection to $20$ divisions,the total resistance in series should be: (in $Omega$)

$\int \left( \frac{2 - \sin 2x}{1 - \cos 2x} \right) e^x \, dx$ is equal to

In the circuit shown,a potential difference of $60\,V$ is applied across $AB$. The potential difference between the points $M$ and $N$ is.....$V$