And accelerated across a gap to the anode, which is held at...

and accelerated across a gap to the anode, which is held at positive potential $V_{0} .$ The cloud of moving electrons within the gap (called space charge) quickly builds up to the point where it reduces the field at the surface of the cathode to zero. From then on a steady current I flows between the plates.Suppose the plates are large relative to the separation $\left(A \gg d^{2} \text { in Fig. } 2.55\right)$, so that edge effects can be neglected. Then $V, \rho,$ and $v$ (the speed of the electrons) are all functions of $x$ alone. (a) Write Poisson's equation for the region between the plates. (b) Assuming the electrons start from rest at the cathode, what is their speed at point $x,$ where the potential is $V(x) ?$ (c) In the steady state, $I$ is independent of $x .$ What, then, is the relation between $\rho$ and $v ?$ (d) Use these three results to obtain a differential equation for $V$, by eliminating $\rho$ and $v$. (e) Solve this equation for $V$ as a function of $x, V_{0},$ and $d .$ Plot $V(x),$ and compare it to the potential without space-charge. Also, find $\rho$ and $v$ as functions of $x$. (f) Show that $$I=K V_{0}^{3 / 2},$$ and find the constant $K$. (Equation 2.56 is called the Child-Langmuir law. It holds for other geometries as well, whenever space-charge limits the current. Notice that the space-charge limited diode is nonlinear-it does not obey Ohm's law.)

and accelerated across a gap to the anode, which is held at positive potential $V_{0} .$ The cloud of moving electrons within the gap (called space charge) quickly builds up to the point where it reduces the field at the surface of the cathode to zero. From then on a steady current I flows between the plates.Suppose the plates are large relative to the separation $\left(A \gg d^{2} \text { in Fig. } 2.55\right)$, so that edge effects can be neglected. Then $V, \rho,$ and $v$ (the speed of the electrons) are all functions of $x$ alone.
(a) Write Poisson's equation for the region between the plates.
(b) Assuming the electrons start from rest at the cathode, what is their speed at point $x,$ where the potential is $V(x) ?$
(c) In the steady state, $I$ is independent of $x .$ What, then, is the relation between $\rho$ and $v ?$
(d) Use these three results to obtain a differential equation for $V$, by eliminating $\rho$ and $v$.
(e) Solve this equation for $V$ as a function of $x, V_{0},$ and $d .$ Plot $V(x),$ and compare it to the potential without space-charge. Also, find $\rho$ and $v$ as functions of $x$.
(f) Show that
$$I=K V_{0}^{3 / 2},$$
and find the constant $K$. (Equation 2.56 is called the Child-Langmuir law. It holds for other geometries as well, whenever space-charge limits the current. Notice that the space-charge limited diode is nonlinear-it does not obey Ohm's law.)

Key Concepts

Poisson's Equation

Poisson's equation relates the second derivative of the electric potential to the local charge density. In the context of vacuum diodes or space-charge problems, it is used to determine how the potential varies throughout the region between the electrodes, accounting for the buildup of charge that alters the electric field distribution.

Energy Conservation in Electron Acceleration

By applying energy conservation to electrons that are accelerated through a potential difference, one can relate the potential at a given point to the kinetic energy (and thus the speed) of the electrons. This concept is fundamental when determining how the electrons speed up from rest as they move from the cathode to the anode.

Steady-State Current Continuity

In a steady-state condition, the electric current must be constant throughout the region between the electrodes. This requirement links the charge density of the moving electrons to their velocity, ensuring that the same amount of charge passes through any cross-section per unit time, a direct consequence of the conservation of charge.

Differential Equations for Potential Distribution

Combining Poisson's equation with the relations obtained from energy conservation and current continuity yields a differential equation solely in terms of the electric potential. Solving this equation provides a self-consistent description of how the potential varies in space in the presence of space charge.

Child-Langmuir Law

The Child-Langmuir law describes the maximum current that can be derived from a vacuum diode under space-charge limited conditions, showing that the current scales as the three-halves power of the applied voltage. It is a classic result in the study of electron flow in vacuum electronics, highlighting the non-linear relationship between the current and the voltage due to space-charge effects.

Space-Charge Effects

Space-charge effects arise from the accumulation of charge in a region, which in turn modifies the local electric field. In vacuum diodes, the buildup of electrons in the gap leads to a self-limiting effect on electron emission and current flow, and a proper understanding of these effects is crucial for accurately modeling and designing devices where electron dynamics play a key role.

Transcript

00:01 For this problem on the topic of electrostatics, we have a vacuum diode where electrons are boiled off a hot cathode at potential zero and then accelerated across the gap to the anode.

00:12 This is held at a positive potential v0 and a cloud of moving electrons within the gap quickly builds to a point where it reduces the field at the surface of the cathode to zero.

00:22 We then have a steady current flowing between the plates.

00:27 And if we suppose that the plates are large relative to the separation so that the edge of x can be neglected then we are asked to find poisson's equation for the region between the plates and then find the speed of the electrons at point x where the potential is v of x we then want to find the relationship between row and v and then use these results to obtain an differential equation for v by eliminating in row and v.

00:54 We then want to solve this equation for v as a function of x, and then plot v of x and compare to the potential without the space charge.

01:02 And lastly, we want to show that i is equal to k times v0 of the power 3 over 2.

01:08 So for a, we have nabler squared of v is equal to minus row over epsilon not, which is poisson's equation.

01:23 And so this is simply d2v, d x squared equal to minus 1 over epsilon 0 times the charge distribution row.

01:40 For part b, we have q times v equal to a half mv squared.

01:49 So the electric potential energy is equal to the kinetic energy and rearranging, we find the speed of the electrons v to be the square root of 2q times the potential big v over the mass of the particles m.

02:09 For part c we have infinitesimal charge unit dq equal to the area a times the charge density row times dx which we can write as a times row times dx d t which is the area a times row times v, which is equal to the current, which is constant.

02:46 For part d, we have the second derivative of the potential with respect to x, d2v, the x squared, equal to minus 1 over epsilon knot times row, which is minus 1 over epsilon 0 times i over av.

03:05 And this is equal to minus i over epsilon not a times the square root of m divided by 2 q v therefore implies that d2v d x squared is equal to beta v to the minus a half where we have beta here to be minus i over epsilon naught a times the square root of m over 2 q...

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