Question

A long Iron slab of width w and height h emerges from a furnace, as shown in Figure 19.79. Because the end of the slab near the furnace is hot and the other end Is cold, the electron mobility increases significantly with the distance x. The electron mobility is$\mathit{u}\mathbf{=}{\mathit{u}}_{\mathbf{0}}\mathbf{+}\mathit{k}\mathit{x}$where u₀is the mobility of the iron at the hot end of the slab. There are n iron atoms per cubic meter, and each atom contributes one electron to the sea of the mobile electron (we can neglect the small thermal expansion of the iron). A steady state conventional current runs through the slab from the hot end towards cold end, and an ammeter (not shown) measures the current to have a magnitude I in amperes. A voltmeter is connected to two locations a distance d apart, as shown. (a) Show the electric field inside the slab at two locations marked with$\mathbf{\times }$. Pay attention to the relative magnitudes of the two vectors that you draw. (b) Explain why the magnitude of the electric field is different at these two locations. (c) At a distance x from the left voltmeter connection, what is the magnitude of the electric field in terms x and the given quantities w, h, d, u₀, k, l, and n ( and fundamental constants)? (d) What is the sign of potential difference displayed on the voltmeter? Explain briefly. (e) In terms of the given quantitiesw, h, d, u₀, k, l, and n and ( and fundamental constants), what is the magnitude of the voltmeter reading? Check your work. (f) What is the resistance of this length of the iron slab?

Short answer

(b) The electric field will decrease and will have two different magnitudes at two locations that are marked as the mobility is increasing, and these two are inversely proportional to each other.

Step-by-step solution

Given data

A long Iron slab with a width w and height h and has temperature difference at both ends. There are n number of iron atoms in a cubic meter.

The electron mobility can be determined by the equation, $u=u_0+kx$ where $u_0$is the mobility of the iron at the hot end of the slab, and it is increasing with distance x.

The steady-state current is denoted as I, which runs through the slab. A voltmeter is connected to measure the voltage between distance d.

Concept

The number of charge carriers traveling in a conductor can be written as,

${\mathit{n}}_{\mathbf{e}}\mathbf{=}\mathit{n}\mathit{A}\mathit{u}\mathit{E}\mathit{t}$ (1)

Here n is the charge density of the charge carrier, A is the cross-sectional area of the conductor, E is the electric field in the conductor, u is the mobility of the charge carriers, t is the time.

(b)  Reason for the different magnitude of the electric field at two locations

The above equation tells us that the electric field and mobility are inversely proportional, and mobility is increasing with distance x in this case which will lead to a decrease in the magnitude of the electric field.

Thus, the electric field will decrease in the direction when mobility is increasing in the same direction and will have two different magnitudes at two locations that are marked.