A $1 \mu \mathrm{C}$ point charge is held at the origin of a cartesian coordinate system. If a second point charge of $10 \mu \mathrm{C}$ is moved from $(0,10,0)$ to $(5,5,5)$ and subsequently to $(5,0,0)$, then the total work done is

$\_\_\_\_$ mJ . (Round off to 2 decimal places). Take $\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9$ in SI units. All coordinates are in meters.

One Coulomb of point charge moving with a uniform velocity $10 \hat{x} \mathrm{~m} / \mathrm{s}$ enters the region $x \geq 0$ having a magnetic flux density $\vec{B}=(10 y \hat{x}+10 x \hat{y}+10 \hat{z}) \mathrm{T}$. The magnitude of force on the charge at $x=0^{+}$is

$\_\_\_\_$ N. ( $\hat{x}, \hat{y}$ and $\hat{z}$ are unit vectors along x -axis, y -axis and z -axis respectively)

Consider a large parallel plate capacitor. The gap $d$ between the two plates is filled entirely with a dielectric slab of relative permittivity 5 . The plates are initially charged to a potential difference of V volts and then disconnected from the source. If the dielectric slab is pulled out completely, then the ratio of the new electric field $E_2$ in the gap to the original electric field $E_1$ is $\_\_\_\_$ .

Let $p$ and $q$ be real numbers such that $p^2+q^2=1$. The eigen values of the matrix $\left[\begin{array}{cc}p & q \\ q & -p\end{array}\right]$ are