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 ×. 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

(d) The voltmeter will display a negative sign.

Step-by-step solution

Given data

The given data can be listed below as,

• A long Iron slab has a width, w, and a height, h, which has One hot end and one cold end.
• There are n iron atoms given in per cubic meter.
• The electron mobility is given as $u=u_0+kx$where role="math" localid="1662999782973" $u_0$is the mobility of the iron at the hot end of the slab, which increases as we move away from the left end.
• The steady-state current is I. A voltmeter is connected to measure the potential drop between two locations that are at a finite distance d.

Concept

A voltmeter is used in an electric circuit to measure the potential drop of potential difference between two points from where its terminals are connected. If the positive terminal is connected to the point that has a higher potential than the other point which is connected to the negative terminal, it will show a positive value.

(d) Determining  the sign of potential difference displayed on the voltmeter

We have calculated that the electric field decreases from left to right, and we know that voltage increases in the opposite direction of the electric field. (refer to SID: 875865-19-61 P-b)

The positive terminal is connected to the cold end, which has a lower potential, and the negative terminal is connected to the hot end, which is a higher potential. Because of this setup, it will show a negative sign.

Thus, the voltmeter will display a negative sign.