Q$1$: Transform the vector field, vec$(A)$ from the spherical coordinate system into the cylindrical and cartesian coordinate systems

vec$(A)=2cos()vec(a_r)+3sin()vec(a_{})+2rcos()vec(a_{})$

Q$2$: Assume the vector field function vec$(A).$ Calculate $o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vec$(A)*vec(dl)$ closed contour integral with reference to the Figure.

vec$(A)=[{}^{2}co{s}^2()sin()(3-sin())]vec(a_{})-[{}^{2}sin()cos()(3sin()+co{s}^2())]vec(a_{})$

Q$3$: Three point charges of different magnitudes $Q_1=2Q_2=-3Q_3$ are located at three corner points of equilateral triangle with edge length of $L$ as shown in Figure. Calculate the total electric field resulting from these three charges at the triangle center point in terms of Q$1.$

Q$4$: $($a$)$ State Gauss Law and calculate the electric field in each of threc icyivio, pivuuceu vy a

$(r)={}_0[1-\frac{r^2}{b^2}]$

spherical distribution of charge

$$ charge distribution is concentrically surrounded by existing in the region $0rb,$ where the outer radius,r$0.$

Q$5$: A total charge of Q$1$ is distributed uniformly along a half$-$circular ring as shown in Figure. Two point charges, each of magnitude Q$2$ are situated as shown. The surrounding medium is free space. Find Q$2$ in terms of Q$1$ so that the electric field at the center of the ring is zero.