A vacuum tube diode (see Exercise 23.27) consists of con-...

A vacuum tube diode (see Exercise 23.27$)$ consists of con- centric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is not a linear function of the position, even with planar geometry, but is given by $$ V(x)=C x^{4 / 3} $$ where $x$ is the distance from the cathode and $C$ is a constant, characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is 13.0 $\mathrm{mm}$ and the potential difference between electrodes is 240 $\mathrm{V}$ . (a) Determine the value of $C .$ ( b) Obtain a formula for the electric field between the electrodes as a function of $x$ . (c) Determine the force on an electron when the electron is halfway between the electrodes.

Key Concepts

Nonlinear Electric Potential Distribution

This concept involves understanding that the potential in certain systems, such as vacuum tube diodes where space-charge effects are significant, does not vary linearly with distance. Instead, the potential takes on a non-linear form (for example, V(x)=C x^(4/3)), which reflects the influence of charge accumulation and other physical phenomena on the spatial distribution of voltage.

Relationship Between Electric Potential and Electric Field

A fundamental principle in electromagnetism is that the electric field is the negative derivative of the electric potential with respect to position. This concept is critical for deriving the electric field from a known potential function, allowing one to relate how the spatial change in potential drives the electric force experienced by charges.

Use of Boundary Conditions to Determine Constants

In many physical problems, unknown constants in mathematical expressions for potential or field distributions are determined by applying boundary conditions. This concept involves using known values, such as the potential difference across a given distance, to solve for constants and ensure that the model accurately reflects the physical setup.

Force on a Charged Particle in an Electric Field

This concept is centered on the principle that a charged particle experiences a force when placed in an electric field, given by F = qE. Understanding this relationship allows one to compute the force on particles, such as electrons, by evaluating the electric field at the particle's location and multiplying by its charge.

Transcript

Please give Ace some feedback