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

(f) The resistance of this length of the iron slab is $\frac{1}{newhk}\left(\log \left(\frac{u_0+kd}{u_0}\right)\right)$.

Step-by-step solution

Given data

The given data can be listed below as,

• A long Iron slab with w width and h height has a hot and cold end.
• n is the number of iron atoms per cubic meter which is given.
• The electron mobility is given with the help of the equation $u=u_0+kx$, in which$u_0$ is given as the mobility of the iron at the hot end of the slab, and the mobility increases as we move away from the left end with respect to distance x.
• The steady-state current that is flowing in the slab is given as I. A voltmeter is there, which is connected to the circuit to measure the potential drop between two locations that are at a finite distance d.

Concept

Ohms's law provides us with the relation between the flowing current I in the conductor and the voltage V between two ends of the conductor when we know the resistance R of the conductor.

The formula is given as,

$\mathit{R}\mathbf{=}\frac{\mathit{V}}{\mathit{I}}$

(f) Calculation of the resistance

The magnitude of the voltage across this length is $\frac{I}{newhk}\left(\log \left(\frac{u_0+kd}{u_0}\right)\right)$. (refer to sid: 875865-19-61 P-e)

Substitute the value in the above equation, and we get,

$\begin{array}{rcl}R& =& \frac{\frac{I}{newhk}\left(\log \left(\frac{u_0+kd}{u_0}\right)\right)}{I}\\ & =& \frac{1}{newhk}\left(\log \left(\frac{u_0+kd}{u_0}\right)\right)\\ & & \end{array}$

Thus, the resistance of this length of the iron slab is $\frac{1}{newhk}\left(\log \left(\frac{u_0+kd}{u_0}\right)\right)$.