Problem $1$

Two infinite wires carry current in the hat$(z)$ direction have position vectors in the xy$-$plane of vec$(r{)}_1=(-a,-b)$ and vec$(r{)}_2=(a,-b)$ with corresponding currents $i_1$ and $i_2,$ respectively. Simultaneously, two charges, $q_3$ and $q_4,$ are located at vec$(r{)}_3=(a,b)$ and vec$(r{)}_4=(-a,b),$ respectively.

For a particle with charge $q$ and velocity vector vec$(v)=(v_x,v_y,v_z)$ determine the net force on the particle at the origin.

Problem $2$

A circular loop of wire of radius $r$ has a normal vector of hat$(n)=(0,0,1)$ at time $t=0$ initially before it begins rotating in uniform circular motion in the yz$-$plane with a period of $T$ seconds. A magnetic field vec$(B)=B(b_x,b_y,b_z)$ is present as the loop rotates.

If the loop is connected to an electronic circuit that can be modeled as a resistor with resistance, $R,$ determine the power delivered to the circuit as a function of time.