Why can we just write $\cdot [\vec u_RdR+\vec u_\theta Rd\theta] $ to describe this irregular path?

Question

1. Calculate the work that charge $q_O$ do to $q_t$ when $q_t$ moves from $P_1$ to $P_2$ in the first picture, and here is its formula

$$W=\int^{P_1}_{P_2}\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2}\cdot[\vec u_RdR+\vec u_\theta Rd\theta]$$

I want to ask why can we just write $\cdot [\vec u_RdR+\vec u_\theta Rd\theta] $ to describe this irregular path ?

In the second picture,we have to calculate the work that charge $q_O$ do to $q_t$ when $q_t$ moves from $P_1$ to $P_2$ along the red line path, the radius of inner circle and outer circle is $r_1$ and $r_2$ and here is its formula

$$W=W_1+W_2$$

$$W_1=\int^{r_2}_{r_1}\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2} \cdot \vec u_R dR$$

$$W_2=\int^{\frac{\theta}{2}}_{\theta=0}\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2} \cdot \vec u_\theta Rd\theta$$

in the second picture,i want ask why should write

1.$\vec u_R \frac{q_oq_t}{4\pi\epsilon_{0}R^2}$, not $\vec u_\theta \frac{q_oq_t}{4\pi\epsilon_{0}R^2}$,why must the direction be R,not $\theta$ ?

2.why should we $\cdot \vec u_\theta Rd\theta$ in the $W_2$, instead of $\cdot \vec u_R Rd\theta$

Answer 1

Half of your integral is irrelevant since you take the dot product between perpendicular vectors $$W =\int \vec{F} \cdot \vec{dR}$$For the electric field $$W = \int|F|\hat r \cdot \vec{dR} $$ the position vector in spherical coordinates is $$\vec{R} = r \hat r$$Using the product rule $$ \frac{d\vec{R}}{dt} = r \frac{d\hat r}{dt} + \frac{dr}{dt} \hat r$$ the first expression can be expressed in terms of $\hat \theta ,\hat \phi$ , multiply by dt to find dr, what remains is just $\int |F| dr$ since the dot product with the electric field unit vector and the other unit vectors are zero by definition