The current flowing along the path $A B C D$ of a cube (shown in the left figure) produces a magnetic field at the centre of the cube of magnitude $B$. Dashed lines depict the non-conducting part of the cube. Consider a cubical shape shown to the right which is identical in size and shape to the left. If the same current now flows in along the path $D A E F G C D$,then the magnitude of the magnetic field at the centre will be

$A$ conductor lies parallel to the $Z$-axis between $-1.5 \le Z < 1.5 \text{ m}$,carrying a constant current of $10.0 \text{ A}$ in the $-\hat{a}_z$ direction. For the given magnetic field $\vec{B} = 3.0 \times 10^{-4} e^{-0.2x} \hat{a}_y \text{ T}$,find the power required to move the conductor at a constant speed from $x = 0$ to $x = 2.0 \text{ m}$ in a time interval of $5 \times 10^{-3} \text{ s}$. Assume the motion is parallel to the $X$-axis. ........... $\text{W}$

In a Thomson mass spectrograph,the electric field and magnetic field are applied:

$A$ charged particle of mass $m$ and charge $q$ moving under the influence of a uniform electric field $E\hat{i}$ and a uniform magnetic field $B\hat{k}$ follows a trajectory from point $P$ to $Q$ as shown in the figure. The velocities at $P$ and $Q$ are respectively $v\hat{i}$ and $-2v\hat{j}$. Which of the following statements $(A, B, C, D)$ are correct? (Trajectory shown is schematic and not to scale)
$(A)$ $E = \frac{3}{4}\left(\frac{mv^{2}}{qa}\right)$
$(B)$ Rate of work done by the electric field at $P$ is $\frac{3}{4}\left(\frac{mv^{3}}{a}\right)$
$(C)$ Rate of work done by both the fields at $Q$ is zero
$(D)$ The difference between the magnitude of angular momentum of the particle at $P$ and $Q$ is $2mav$.

$A$ compass needle free to turn in a horizontal plane is placed at the centre of a circular coil of $30$ turns and radius $12 \;cm$. The coil is in a vertical plane making an angle of $45^{\circ}$ with the magnetic meridian. When the current in the coil is $0.35 \;A$,the needle points west to east.
$(a)$ Determine the horizontal component of the earth's magnetic field at the location.
$(b)$ The current in the coil is reversed,and the coil is rotated about its vertical axis by an angle of $90^{\circ}$ in the anticlockwise sense looking from above. Predict the direction of the needle. Take the magnetic declination at the place to be zero.

One Tesla is equal to