Chapter 19: Q61P-a (page 800)

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 $u = u_{0} + k x$ where $u_{0}$ is 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 $\times$ . 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 quantities w,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

Expert verified

At both the cross symbol, the direction of the electric field is from left to right.

Step by step solution

Given data

A long Iron slab has a width w and height of h and emerges from a furnace. The symbol n denotes the iron atoms per cubic meter. One end is hot, and the other one is cold. The electron mobility is $u = u_{0} + k x$ where $u_{0}$ is the mobility of the iron at the hot end of the slab, and it increases with distance x. The steady-state current is I, which runs through the slab from the hot end toward the cold end. A voltmeter is connected to measure the voltage between two points which are at a distance d.

Concept

The electric field is generated by the charge carriers when they are present in the space. The other charge, which is moving from a finite to an area, will feel the field if they are in the electric field generated by a charge which was already present in the area.

(a) The electric field inside the slab at two locations marked with ×

The left end of the slab is hot and connected to the negative terminal of the voltmeter, and the other end is cold and connected to the positive terminal of the voltmeter.

The electron will move from the hot end to the cold end the direction of the electric field will be left to right. This can be shown as,