Distance between electrodes and depth of investigation In geophysics and specifically electrical/resistivity survey methods there is a rule of thumb, which states that the longer is the distance between source electrodes, the deeper is investigation depth. Although, I might kind of feel the intuition of this phenomenon, but I can’t get my head around the concrete proof and mathematical/physical explanation for this effect. The diagram talking about what I described is below:

 A: Well this is the first time I hear about this method and I am not sure how it really works and what data you get from it, but I a vague idea that you would want to know the approximate resistivity of the ground below and use that for some kind of analysis.
In that case you would want to pass current through the ground by using source above to provide electrodes with voltage. They would then create electric field resembling the red lines shown in your diagrams. Now soil is obviously not a very good conductor but at high enough voltages and sensitive enough equipment, I imagine you would be able to pass a small but measurable current through the ground and use it to determine the resistance down below.
Why the distance between the electrodes matter?
Short answer - if the electrodes are too close then most of the current will flow in a straight line from one electrode to the other, so the harder it would get to collect any information of the resistivity of the layers below.
To show that let's assume the ground is a homogeneous material with resistivity $\rho$ and that air is a perfect insulator and that the electrodes are point-like charges $+q$ and $-q$, such that electrons can flow from/to them, but they remain with almost constant charge.
(Obviously, $\rho$ being constant doesn't make much sense here, given we are interested in surveying its distribution in the ground, but as you'll see we can easily make it a function of depth etc. but it is easier for now to leave it as a constant)

Consider the image from above. $A$ and $B$ are the electrodes and we are trying to find current flowing through the black circle of radius $R$ which is half way between the electrodes as shown. We said that we take $\rho_{air} = \infty$ therefore we consider current flowing only in the lower half space.
We can easily find E-field in points which lie on the lower half of the disk shown in the image. You can verify that $\vec E$ is always orthogonal to the disk, pointing from $+$ to $-$ and
$$\vert \vec E \vert = \frac{2qd}{4\pi\varepsilon}\cdot \frac{1}{\left( r^2+d^2 \right)^{\frac32}}$$
where $r$ is distance of point on disk from origin, and $d$ is distance of each electrode from origin.
After that we can find current flowing through this disk:
$$I_{disk} = \int_{disk} \vec j \cdot d\vec S = \int_{disk} \frac1{\rho} \vec E \cdot d\vec S \\ = \int_{disk} \frac1\rho E \cdot dS = \int_0^\pi d\varphi \int_0^R \frac{qd}{2\pi\varepsilon\rho} \cdot \frac{rdr}{\left( r^2+d^2 \right)^\frac32} = \frac{qd}{2\rho\varepsilon} \left( \frac1d - \frac1{\sqrt{R^2+d^2}} \right)$$
If we let disk radius $R \to \infty$, then we get current through the whole conductive part of the plane situated half way between the electrodes.
$$I_{total} = \frac q{2\rho\varepsilon}$$
$I_{total}$ notably doesn't depend on $d$. And here is the graph of $\frac{I_{disk}}{I_{total}}$ as a function of $d$.

As you increase distance between the electrodes, the current tends to get weaker through a disk of radius $R$, but since we know that the total current remains the same, it follows that bigger part of total current runs deeper and wider, thus enabling us to collect more information about deeper layer of the ground.
This should not be considered a proof but rather hand-waving that is helpful in understanding the phenomenon. Also, in real world things get much more complicated very quickly as you start removing rough approximations used here.
EDIT:
The term penetration depth is used in geology, but its meaning is a matter of definition. A tiny current always penetrates to infinite depth. But we can define a certain threshold. For example we can define penetration depth to be the depth $R$ such that $50\%$ of all current flows above $R$. Through the prism of the text above it would mean that $I_{disk} = \frac12 I_{total}$
If you remember, $I_{disk}$ does not only depend of distance $d$ between electrodes, but also $R$ which is the radius of the disk (I'll emphasize it like this $I_{disk}^R (d)$). If I decide to define penetration depth as I did here, that would mean that $R$ would be penetration depth if $I_{disk}^R(d) = \frac12 I_{total}$ i.e.
$$R \ \text{is penetration depth} \iff \frac{I_{disk}^R (d)}{I_{total}} = \frac12$$
So function that I plotted above, also has $R$ as a parameter, and as you increase it, so will increase $d$ for which $R$ satisfies the definition of penetration depth.
Once again, plot of $\frac{I_{disk}^R}{I_{total}}(d) = 1-\frac{d}{\sqrt{R^2+d^2}}$ for different values of $R$:


If you increase penetration depth, you must also increase $d$ to achieve that depth, as can be seen on the graphs.
You can play with this function here
