I'm having trouble following an example in the book. I don't understand a few steps which I have marked in bold. Any help would be awesome! Thanks.(adsbygoogle = window.adsbygoogle || []).push({});

Q: In vacuum-tube diodes, electrons are emitted from a hot cathode at zero potential and collected by an anode maintained at a potential [itex] V_0 [/itex], resulting in a convection current flow. Assuming that the cathode and the anode are parallel conducting plates and that hte electrons leave the cathode with a zero initial velocity (spache-charge limited condition), find the relation between teh current density [itex] \vec C [/itex] and [itex] V_0 [/itex].

A:

Neglecting fringing effects we have,

[tex] \vec E(0) = \vec a_y E_y(0) = -\vec a_y \frac{dV(y=0)}{dy} = 0[/tex]

In the steady state the current density is constant, independent of y:

[tex] \vec J = -\vec a_y J = \vec a_y \rho(y) u(y) [/tex]

where the charge density [itex] \rho (y) [/itex] is a negative quanitity. The velocity [itex] \vec u = \vec a_y u(y) [/itex] is related to the electric field intensity [itex] \vec E(y) = \vec a_y E(y) [/itex] by Newton's law of motion:

[tex] m \vec{d u(y)}{dt} = -eE(y) = e \vec{dV(y)}{dy} [/tex], where [itex] m [/itex] and [itex] e [/itex] are the mass and charge respectively of an electron. Noting that:

This is where I am confused. I am not noting anything =)

[tex] m \frac{du}{dt} = m \frac{du}{dy} \frac{dy}{dt} = mu \frac{du}{dy} [/tex]

[tex] = \frac{d}{dy} \left( \frac{1}{2} mu^2 \right) [/tex]

[tex] \frac{d}{dy} \left( \frac{1}{2} m u^2 \right) = e \frac{dV}{dy} [/tex]

I don't understand:

[tex] m \frac{du}{dy} \frac{dy}{dt} \rightarrow mu \frac{du}{dy} [/tex]

[tex] mu \frac{du}{dy} \rightarrow \frac{d}{dy} \left( \frac{1}{2} mu^2 \right) [/tex]

and last but not least...

[tex] \frac{d}{dy} \left( \frac{1}{2} m u^2 \right) \rightarrow e \frac{dV}{dy} [/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Emag - Having trouble following an example

**Physics Forums | Science Articles, Homework Help, Discussion**