At any point the electric field is derivative (rate of change) of electric potemtial with respect to distance.
How does it look like when the cathode is hot?
Is it positive right next to cathode? Does it get negative as electron cloud is approached?
Is it still negative at a distance after the grid?
A nice way to visualize is the use of analog computers operating at the speed of light.
Rubber membranes are stretched, and small balls let to travel. Balls simulate electrons.
Weights hang from the rubber sheet define cathode, anode potential. Many small weights spaced uniformly or not define the potential at the grid wire.
A brilliant drawing of elastic-membrane model showing potential variation within a plane electrode triode with a rectangular weight resting and deforming the membrane for example for the anode, membrane elevation corresponding to negative potential, can be seen on figure 10.2 [Sprangenberg].
It is also nice to hang a weight to a rubber membrane and watch a thrown ball become a satelite round the simulated gravity. Then 2 weights hung and the satelite planet ball orbit may become a complicated one orbiting in chaotic? turns one or the other sun. Then 2 balls can be thrown. Of course it is difficult to simulate the force influence of one ball to the other be it gravitational or electrostatic.
Reference: Fundamentals of electron devices - Karl R. Spangenberg - international student edition - Chapter 10, Control type vacuum tubes - page 205
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