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$\texttt{C O N T E N T S}$
$\texttt{Abstract}$
$\bl\S\texttt{ A. The general case}$
$\bl\S\texttt{ B. Two opposite point electric charges with unequal absolute values}$
$\bl\S\texttt{ C. Two opposite point electric charges with equal absolute values}$
$\bl\S\texttt{ D. Two equal point electric charges}$
$\bl\S\texttt{ E. Two point electric charges of the same sign and unequal values}$
$\bl\S\texttt{ F. Equipotential lines (surfaces)}$
$\bl{=\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!==\!=\!=\!=}$
Abstract
The differential equation obeyed by the electric field lines between two point electric charges $\:q_{\texttt A},q_{\texttt B}\:$ in general is produced. By the solution a set of parametric equations is provided. At first the solution is applied on the special case of the post $\:\plr{q_{\texttt A},q_{\texttt B}}\e \plr{\m q, \p 2q}\:$ and the relevant diagram is created. It is found to be in agreement with diagrams we meet in many textbooks and the Web. Moreover diagrams for other special cases, like $\:\plr{q_{\texttt A},q_{\texttt B}}\e \plr{ q, q}\:$ or $\:\plr{q_{\texttt A},q_{\texttt B}}\e \plr{q, 3q}\:$, are created.
$\bl{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}$
$\bl\S$ A. The general case
Two point electric charges $\:q_{\texttt A},q_{\texttt B}\:$ are standing still on the $\:\mr x\m$axis of a system $\:\mr{Oxyz}\:$ by a distance $\:2a\:$ apart, see Figure-01. By the presence of these charges we have a 3-dimensional electrostatic field at every point $\:\texttt P\plr{x,y,z}\:$ represented by the electric field intensity vector $\:\mb E\plr{x,y,z}$. An electric field line is a 3-dimensional curve with the vector $\:\mb E\plr{x,y,z}\:$ tangent to it at every one of its points. Because of the rotational symmetry around the $\:\mr x\m$axis the problem is 2-dimensional, so we will work on the problem in the plane $\:\mr{xy}$.
Suppose that an electric field line is a plane curve with regular parametric representation
\begin{equation}
\mb r\plr{t} \e \plr{\!\!\!\plr{x\plr{t},y\plr{t}\Vp{A^B_C}}\!\!\!}
\tl{A-01}
\end{equation}
the variable $\:t\:$ being the parameter of the representation. A vector tangent to it at a point $\:\texttt P\plr{x,y}\:$ is
\begin{equation}
\dfrac{\mr d\mb r}{\mr dt}\e \plr{\dfrac{\mr dx}{\mr dt},\dfrac{\mr dy}{\mr dt}\Vp{A^B_C}}
\tl{A-02}
\end{equation}
Since the electric field intensity vector $\:\mb E\plr{x,y}\e\plr{\!\!\!\plr{E_x\plr{x,y},E_y\plr{x,y}\Vp{A^B_C}}\!\!\!} $
is tangent to the curve at this point $\:\texttt P\plr{x,y}\:$ we must have, see Figure-02
\begin{equation}
\dfrac{\mr dy/\mr dt}{\mr dx/\mr dt}\e\dfrac{E_y\plr{x,y}}{E_x\plr{x,y}}
\tl{A-03}
\end{equation}
that is
\begin{equation}
\dfrac{\mr dy}{\mr dx} \e \dfrac{E_y\plr{x,y}}{E_x\plr{x,y}}
\tl{A-04}
\end{equation}
It was proved a posteriori that it would be difficult or may be impossible to succeed separation of the used variables $\:\plr{x,y}\:$ in order to have a solution of this differential equation. This equation was solved by an accidentally successful change of variables. The new variables are
\begin{equation}
u \e \tan\theta_{\mr A} \,,\qquad v \e \tan\theta_{\mr B}\qquad \text{with} \quad \theta_{\mr A},\theta_{\mr B}\bl\in \blr{0,2\pi}
\tl{A-05}
\end{equation}
where $\:\plr{\theta_{\mr A},\theta_{\mr B}}\:$ the angles shown in Figure-01 or Figure-02.
Using the geometry of the problem we'll transform the differential equation \eqref{A-04} with respect to $\:\plr{x,y}\:$ to a differential equation with respect to $\:\plr{ u, v}$.
At first we have
\begin{equation}
\plr{x \m a}\tan\theta_{\mr B} \e y \e \plr{x \p a}\tan\theta_{\mr A}
\nonumber
\end{equation}
or
\begin{equation}
\plr{x \m a} v \e y \e \plr{x \p a} u
\tl{A-06}
\end{equation}
so
\begin{equation}
x \e \dfrac{\tan\theta_{\mr B}\p\tan\theta_{\mr A}}{\tan\theta_{\mr B}\m\tan\theta_{\mr A}}\, a \,,\qquad y \e \dfrac{2\tan\theta_{\mr B} \tan\theta_{\mr A}}{\tan\theta_{\mr B} \m \tan\theta_{\mr A}}\, a
\tl{A-07}
\end{equation}
or
\begin{equation}
x \e \plr{\dfrac{v\p u}{v\m u}} a \,,\qquad y \e \plr{\dfrac{2\,v\, u}{v \m u}} a
\tl{A-08}
\end{equation}
From \eqref{A-08}
\begin{equation}
\mr dx \e 2\,\dfrac{v\mr du\m u\mr d v}{\plr{v\m u}^2}\, a \,,\qquad \mr dy \e 2\,\dfrac{v^2\mr du\m u^2\mr d v}{\plr{v\m u}^2}\, a
\tl{A-09}
\end{equation}
From \eqref{A-09} the left hand side of \eqref{A-04} is transformed as follows
\begin{equation}
\dfrac{\mr dy}{\mr dx} \e \dfrac{v^2\mr du\m u^2\mr d v}{v\mr du\m u\mr d v}
\tl{A-10}
\end{equation}
For the transformation of the right hand side of \eqref{A-04} we have at first
\begin{align}
E_x & \e k\plr{q_{\mr A}\dfrac{x \p a}{\vlr{\mr{PA}}^3}\p q_{\mr B}\dfrac{x \m a}{\vlr{\mr{PB}}^3}}
\tl{A-11a}\\
E_y & \e k\plr{q_{\mr A}\dfrac{y}{\,\vlr{\mr{PA}}^3}\p q_{\mr B}\dfrac{y}{\,\vlr{\mr{PB}}^3}\Vp{\dfrac{x \p a}{\mr{PA}^3}}}
\tl{A-11b}
\end{align}
so
\begin{equation}
\dfrac{E_y}{E_x} \e \dfrac{\plr{\dfrac{q_{\mr A}}{\,\vlr{\mr{PA}}^3}\p \dfrac{q_{\mr B}}{\,\vlr{\mr{PB}}^3}\Vp{\dfrac{x \p a}{\mr{PA}^3}}}y}{\plr{\dfrac{q_{\mr A}}{\,\vlr{\mr{PA}}^3}\p \dfrac{q_{\mr B}}{\,\vlr{\mr{PB}}^3}\Vp{\dfrac{x \p a}{\mr{PA}^3}}}x\p \plr{\dfrac{q_{\mr A}}{\,\vlr{\mr{PA}}^3}\m \dfrac{q_{\mr B}}{\,\vlr{\mr{PB}}^3}\Vp{\dfrac{x \p a}{\mr{PA}^3}}}a}
\tl{A-12}
\end{equation}
From the geometry of the problem
\begin{equation}
\vlr{\mr{PA}} \e \dfrac{y}{\sin\theta_{\mr A}} \,,\qquad \vlr{\mr{PB}} \e \dfrac{y}{\sin\theta_{\mr B}}
\tl{A-13}
\end{equation}
and \eqref{A-12} yields
\begin{equation}
\dfrac{E_y}{E_x} \e \dfrac{\plr{q_{\mr A}\sin^3\theta_{\mr A} \p q_{\mr B}\sin^3\theta_{\mr B}}y}{\plr{q_{\mr A}\sin^3\theta_{\mr A} \p q_{\mr B}\sin^3\theta_{\mr B}}x \p \plr{q_{\mr A}\sin^3\theta_{\mr A} \m q_{\mr B}\sin^3\theta_{\mr B}}a}
\tl{A-14}
\end{equation}
Replacing $\:x,y\:$ by their expressions \eqref{A-07} in terms of
$\:\tan\theta_{\mr A},\tan\theta_{\mr B}\:$ we have
\begin{equation}
\dfrac{E_y}{E_x} \e \dfrac{q_{\mr A}\sin^3\theta_{\mr A} \p q_{\mr B}\sin^3\theta_{\mr B}}{ q_{\mr A}\sin^2\theta_{\mr A}\cos\theta_{\mr A}\p q_{\mr B}\sin^2\theta_{\mr B}\cos\theta_{\mr B}}
\tl{A-15}
\end{equation}
Expressing the trigonometric functions $\:\sin\theta_{\mr A},\cos\theta_{\mr A}\:$ and $\:\sin\theta_{\mr B},\cos\theta_{\mr B}\:$ in terms of the new variables $\:\tan\theta_{\mr A} \e u\:$ and $\:\tan\theta_{\mr B} \e v\:$ respectively, that is
\begin{align}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sin\theta_{\mr A} & \e \dfrac{\varepsilon_{\mr A}\tan\theta_{\mr A}}{\sqrt{1\p\tan^2\theta_{\mr A}}}\e \dfrac{\varepsilon_{\mr A}u}{\sqrt{1\p u^2}}\,, \quad \cos\theta_{\mr A} \e \dfrac{\varepsilon_{\mr A}}{\sqrt{1\p\tan^2\theta_{\mr A}}}\e \dfrac{\varepsilon_{\mr A}}{\sqrt{1\p u^2}}
\tl{A-16a}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!&\nonumber\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sin\theta_{\mr B} & \e \dfrac{\varepsilon_{\mr B}\tan\theta_{\mr B}}{\sqrt{1\p\tan^2\theta_{\mr B}}}\e \dfrac{\varepsilon_{\mr B}v}{\sqrt{1\p v^2}}\,, \quad \cos\theta_{\mr B} \e \dfrac{\varepsilon_{\mr B}}{\sqrt{1\p\tan^2\theta_{\mr B}}}\e \dfrac{\varepsilon_{\mr B}}{\sqrt{1\p v^2}}
\tl{A-16b}\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!&\nonumber\\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\varepsilon_{\mr A} & \e
\left.
\begin{cases}
\p 1, \quad \texttt{if } \theta_{\mr A}\bl\in \blr{0,\pi/2}\\
\m 1, \quad \texttt{if } \theta_{\mr A}\bl\in \blr{\pi/2, \pi}
\end{cases}
\right\}\,,\quad
\varepsilon_{\mr B} \e
\left.
\begin{cases}
\p 1, \quad \texttt{if } \theta_{\mr B}\bl\in \blr{0,\pi/2}\\
\m 1, \quad \texttt{if } \theta_{\mr B}\bl\in \blr{\pi/2, \pi}
\end{cases}
\right\}
\tl{A-16c}
\end{align}
equation \eqref{A-15} yields
\begin{equation}
\dfrac{E_y}{E_x} \e \dfrac{\varepsilon_{\mr A}q_{\mr A}\plr{1\p v^2}^\frac32 u^3\p\varepsilon_{\mr B}q_{\mr B}\plr{1\p u^2}^\frac32 v^3}{\varepsilon_{\mr A}q_{\mr A}\plr{1\p v^2}^\frac32 u^2\p\varepsilon_{\mr B}q_{\mr B}\plr{1\p u^2}^\frac32 v^2}
\tl{A-17}
\end{equation}
Combining \eqref{A-04}, \eqref{A-10} and \eqref{A-17} we have the differential equation in terms of the new variables $\:u,v$
\begin{equation}
\dfrac{v^2\mr du\m u^2\mr d v}{v\mr du\m u\mr d v}\e\dfrac{\varepsilon_{\mr A}q_{\mr A}\plr{1\p v^2}^\frac32 u^3\p\varepsilon_{\mr B}q_{\mr B}\plr{1\p u^2}^\frac32 v^3}{\varepsilon_{\mr A}q_{\mr A}\plr{1\p v^2}^\frac32 u^2\p\varepsilon_{\mr B}q_{\mr B}\plr{1\p u^2}^\frac32 v^2}
\tl{A-18}
\end{equation}
or
\begin{equation}
\varepsilon_{\mr A}q_{\mr A}\dfrac{u\,\mr du}{\plr{1\p u^2}^\frac32} \e \m\varepsilon_{\mr B}q_{\mr B}\dfrac{v\,\mr dv}{\plr{1\p v^2}^\frac32}
\tl{A-19}
\end{equation}
that is
\begin{equation}
\varepsilon_{\mr A}q_{\mr A}\dfrac{\mr d\plr{1\p u^2}}{\plr{1\p u^2}^\frac32} \e \m\varepsilon_{\mr B}q_{\mr B}\dfrac{\mr d\plr{1\p v^2}}{\plr{1\p v^2}^\frac32}
\tl{A-20}
\end{equation}
which integrated gives the solution
\begin{equation}
q_{\mr A}\dfrac{\varepsilon_{\mr A}}{\sqrt{1\p u^2}} \e \m q_{\mr B}\dfrac{\varepsilon_{\mr B}}{\sqrt{1\p v^2}}\p \texttt{constant}
\tl{A-21}
\end{equation}
that is
\begin{equation}
q_{\mr A}\cos\theta_{\mr A} \p q_{\mr B}\cos\theta_{\mr B} \e \texttt{constant}
\tl{A-22}
\end{equation}
We express this equation in the form
\begin{equation}
\boxed{\:\:\cos\theta_{\mr A} \p \lambda_q\cos\theta_{\mr B} \e \mr k \e \text{constant}\:\:\Vp{\tfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}}
\tl{A-23a}
\end{equation}
where
\begin{equation}
\boxed{\:\: \lambda_q \stackrel{\mr{def}}{\bl\equiv} \dfrac{q_{\mr B}}{q_{\mr A}}\e \text{ratio of the electric charges}\:\:\vp}
\tl{A-23b}
\end{equation}
Anyone of the electric field lines is created by \eqref{A-23a} for some constant $\:\mr k$. But not any arbitrary $\:\mr k\:$ corresponds to an
electric field line. From \eqref{A-23a}
\begin{equation}
\mr {k_{max}} \e 1\p \vlr{\lambda_q} \e 1\p \vlr{\dfrac{q_{\mr B}}{q_{\mr A}}}\,,\qquad \mr {k_{min}} \e \m 1\m \vlr{\lambda_q} \e \m 1\m \vlr{\dfrac{q_{\mr B}}{q_{\mr A}}}
\tl{A-24}
\end{equation}
so the entire electric field lines family is created from \eqref{A-23a} with
\begin{equation}
\mr k \bl\in \blr{\m\plr{1\p \vlr{\lambda_q}},\plr{1\p \vlr{\lambda_q}}\vp}\,, \qquad \lambda_q \e \dfrac{q_{\mr B}}{q_{\mr A}}
\tl{A-25}
\end{equation}
To find the parametric equations of the $\:\mr k\m$electric field line we use equations \eqref{A-07}
\begin{align}
x & \e \dfrac{\tan\theta_{\mr B}\p\tan\theta_{\mr A}}{\tan\theta_{\mr B}\m\tan\theta_{\mr A}}\, a
\tl{A-26a}\\
y & \e \dfrac{2\tan\theta_{\mr B} \tan\theta_{\mr A}}{\tan\theta_{\mr B} \m \tan\theta_{\mr A}}\, a
\tl{A-26b}
\end{align}
From \eqref{A-23a}
\begin{equation}
\theta_{\mr A} \e\arccos\plr{\mr k \m \lambda_q\cos\theta_{\mr B}}
\tl{A-27}
\end{equation}
so the $\:{\color{red}{\bl{\theta_{\mr B}}}}\m$parametric equations for the $\:{\color{blue}{\mr k}}\m$electric field line are
\begin{align}
x_{\color{blue}{\mr k}}\plr{{\color{red}{\bl{\theta_{\mr B}}}}} & \e \dfrac{\tan{\color{red}{\bl{{\color{red}{\bl{\theta_{\mr B}}}}}}}\p\tan\blr{\arccos\plr{{\color{blue}{\mr k}} \m \lambda_q\cos{\color{red}{\bl{\theta_{\mr B}}}}}\vp}}{\tan{\color{red}{\bl{{\color{red}{\bl{\theta_{\mr B}}}}}}}\m\tan\blr{\arccos\plr{{\color{blue}{\mr k}} \m \lambda_q\cos{\color{red}{\bl{\theta_{\mr B}}}}}\vp}}\, a
\tl{A-28a}\\
y_{\color{blue}{\mr k}}\plr{{\color{red}{\bl{\theta_{\mr B}}}}} & \e \dfrac{2\tan{\color{red}{\bl{{\color{red}{\bl{\theta_{\mr B}}}}}}}\tan\blr{\arccos\plr{{\color{blue}{\mr k}} \m \lambda_q\cos{\color{red}{\bl{\theta_{\mr B}}}}}\vp}}{\tan{\color{red}{\bl{{\color{red}{\bl{\theta_{\mr B}}}}}}}\m\tan\blr{\arccos\plr{{\color{blue}{\mr k}} \m \lambda_q\cos{\color{red}{\bl{\theta_{\mr B}}}}}\vp}}\, a
\tl{A-28b}\\
{\color{blue}{\mr k}} &\bl\in \blr{\m\plr{1\p \vlr{\lambda_q}},\plr{1\p \vlr{\lambda_q}}\vp}\,, \qquad \lambda_q \e \dfrac{q_{\mr B}}{q_{\mr A}}
\tl{A-28c}
\end{align}
Note that a $\:{\color{blue}{\mr k}}\m$electric field line created by equation \eqref{A-23a} is symmetric with respect to the $\:\mr x\m$axis since if $\:\theta_{\mr A}\:$ and $\:\theta_{\mr B}\:$ satisfy this equation then $\:\theta'_{\mr A}\e 2\pi\m \theta_{\mr A}\:$ and $\:\theta'_{\mr B}\e 2\pi\m \theta_{\mr B}\:$ satisfy it also. This 2-dimensional $\:\mr x\m$mirror symmetry is due to the 3-dimensional rotational symmetry around the $\:\mr x\m$axis.
$\bl{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}$
$\bl\S$ B. Two opposite point electric charges with unequal absolute values
This case corresponds to negative values of the ratio of charges, $\:\lambda_q\les 0\:$, except the value $\:\lambda_q \bl\ne \m 1\:$, that is
\begin{equation}
\lambda_q \e \dfrac{q_{\mr B}}{q_{\mr A}} \bl\in \plr{\m\infty,\m 1} \bl\cup \plr{\m 1,0}
\tl{B-01}
\end{equation}
The range $\:\plr{\m 1,0}\:$ in \eqref{B-01} could be omitted since a case with $\:\lambda_q \bl\in\plr{\m 1,0}\:$ would be considered as case $\:\lambda'_q \e \lambda^{\m 1}_q\bl\in\plr{\m\infty,\m 1}$. Also, for the case $\:\lambda_q \e \m 1\:$ we refer to $\bl\S$ C.
In Figure-03 we have a diagram of the electric field lines for the value $\:\lambda_q\e \m 2$. We'll explain point to point all about the details and numerical values. So according to \eqref{A-25} for the variable $\:\mr k\:$ we have
\begin{equation}
\m 3\leseq\mr k\leseq \p 3
\tl{B-02}
\end{equation}
For these values the $\:\mr k\m$electric field lines are created as follows :
\begin{equation}
\nonumber
\end{equation}
$\:\plr{\m 3\leseq\mr k\les \m 1}\:$ Electric field lines joining the two charges in the upper half of the plane.
$\:\plr{\mr k \e \m 1}\:$ Electric field line with semicircular-like part and straight segment part on the $\:\mr x\m$axis joining the two charges in the upper half of the plane. The semicircular-like part and the straight segment part on the $\:\mr x\m$axis meet at the neutral point $\:\mr{NP}\:$ where $\:\mb E \e \bl 0$. At the neutral point $\:\mr{NP}\:$ we have a discontinuity of the direction of the line tangent to the electric field curve.
$\:\plr{\m 1\les\mr k\les\p 1}\:$ All electric field lines from the positive point charge $\:q_{\mr B}\e \p 2q\:$ to infinity. Especially for $\:\mr k\e 0\:$ the created electric field line moves away to infinity asymptotically to a straight line parallel to $\:\mr y\m$axis which intersects the $\:\mr x\m$axis at point $\:\mr C$. Although we have not a proof it seems that this point is the $^{\prime\prime}$$\textit{electric charge center}$ $^{\prime\prime}$. With reference to Figure-04 we must have
\begin{equation}
x_{\texttt C}\e\dfrac{q_{\texttt A}x_{\texttt A}\p q_{\texttt B}x_{\texttt B}}{q_{\texttt A}\p q_{\texttt B}}
\tl{B-03}
\end{equation}
which applied to Figure-03 gives
\begin{equation}
x_{\texttt C}\e\dfrac{\plr{\m q}\plr{\m a}\p \plr{\p 2q}\plr{\p a}}{\plr{\m q}\p \plr{\p 2q}}\e 3\cdot a \e 3\cdot 2 \e 6
\tl{B-04}
\end{equation}
$\:\plr{\mr k \e \p 1}\:$ Electric field line with semicircular-like part and straight segment part on the $\:\mr x\m$axis joining the two charges in the loewer half of the plane. The semicircular-like part and the straight segment part on the $\:\mr x\m$axis meet at the neutral point $\:\mr{NP}\:$ where $\:\mb E \e \bl 0$. At the neutral point $\:\mr{NP}\:$ we have a discontinuity of the direction of the line tangent to the electric field curve.
$\:\plr{\p 1\les\mr k\leseq \p 3}\:$ Electric field lines joining the two charges in the lower half of the plane.
\begin{equation}
\nonumber
\end{equation}
Note that electric field lines joining the two charges are trapped inside the circular-like region (spherical-like in 3 dimensions) while the electric field lines from the positive point charge $\:q_{\mr B}\e \p 2q\:$ to infinity run outside of this circular-like region.
Figure-04 is identical to Figure-03 but with formulas on the basis of which the numerical values of the latter are derived.
$\bl{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}$
$\bl\S$ C. Two opposite point electric charges with equal absolute values
For this case, where $\:\lambda_q \e \m 1$, see my answer here : Equation describing the electric field lines of opposite charges.
$\bl{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}$
$\bl\S$ D. Two equal point electric charges
For this case, where $\:\lambda_q \e \p 1$, a typical diagram is shown in Figure-05.
$\bl{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}$
$\bl\S$ E. Two point electric charges of the same sign and unequal values
For this case, where $\:\lambda_q \gr\p 1$, a typical diagram is shown in Figure-06. Note that a field line from the absolutely larger charge moves away to infinity asymptotically to a straight line parallel to $\:\mr y\m$axis which intersects $\:\mr x\m$axis at point $\:\mr C$. It seems that this point is the $^{\prime\prime}$$\textit{electric charge center}$ $^{\prime\prime}$. With reference to Figure-06 we must have
\begin{equation}
\dfrac{\vlr{\texttt{CA}}}{\vlr{\texttt{CB}}} \e \dfrac{3q}{q}\e 3
\tl{E-01}
\end{equation}
$\bl{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}$
$\bl\S$ F. Equipotential lines (surfaces)
Electric field lines versus equipotential surfaces (video).
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Watch the production of this image here :
Electric field lines of two unequal opposite charges (Video)
Reference for above Figure : Angle of electrostatic field lines through an equilibrium point, see &Vincent Fraticelli's answer.
