Adding different amounts of impurities. A rod of...

Adding different amounts of impurities. A rod of semiconducting material of length $L$ and cross-sectional area $A$ lies along the $x$ -axis between $x=0$ and $x=L$. The material obeys Ohm's law, and its resistivity varies along the rod according to $\rho(x)=\rho_{0} \exp (-x / L) .$ The end of the rod at $x=0$ is at a potential $V_{0}$ greater than the end at $x=L$. (a) Find the total resistance of the rod and the current in the rod. (b) Find the electric-field magnitude $E(x)$ in the rod as a function of $x$. (c) Find the electric potential $V(x)$ in the rod as a function of $x$. (d) Graph the functions $\rho(x), E(x),$ and $V(x)$ for values of $x$ between $x=0$ and $x=L$.

Adding different amounts of impurities. A rod of semiconducting material of length $L$ and cross-sectional area $A$ lies along the $x$ -axis between $x=0$ and $x=L$. The material obeys Ohm's law, and its resistivity varies along the rod according to $\rho(x)=\rho_{0} \exp (-x / L) .$ The end of the rod at $x=0$ is at a potential $V_{0}$ greater than the end at $x=L$. (a) Find the total resistance of the rod and the current in the rod. (b) Find the electric-field magnitude $E(x)$ in the rod as a function of $x$.
(c) Find the electric potential $V(x)$ in the rod as a function of $x$. (d) Graph the functions $\rho(x), E(x),$ and $V(x)$ for values of $x$ between $x=0$ and $x=L$.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits stating that the voltage across a conductor is proportional to the current through it, with the resistance as the constant of proportionality. In differential form, for materials where properties vary along the dimension, it underlies the relation between the local electric field, current density, and material properties.

Resistance in Non?Uniform Materials

When a material has properties such as resistivity that vary with position, the total resistance must be determined by integrating the differential resistance across the material. This involves dividing the material into infinitesimal segments, calculating the resistance of each, and summing them up to obtain the overall resistance.

Resistivity Variation

Resistivity is an intrinsic property that quantifies how strongly a material opposes the flow of electric current. In cases where resistivity changes with position, its functional form must be considered to evaluate local resistive effects and determine how current and potential vary along the material.

Electric Field in Conductors

The electric field within a conductor is related to the local potential gradient. In an inhomogeneous material, the electric field can change from point to point, and it is calculated as the negative derivative of the electric potential with respect to position. This spatial dependency has important implications for how voltage drops are distributed along the material.

Electric Potential Distribution

The electric potential distribution in a conductor reflects the cumulative effect of the electric field along its length. By integrating the electric field and applying appropriate boundary conditions (such as specified potentials at the ends), one can obtain the potential function, which describes how voltage varies with position.

Graphical Analysis of Physical Quantities

Graphing functions like resistivity, electric field, and potential as a function of position provides visual insight into the behavior of the system. Such representations help in understanding trends, gradients, and spatial variations, making it easier to interpret the underlying physics in non-uniform materials.

Transcript

00:01 So for part a, we have dr is equal to dx over a, which is equal to p -not, x -p -negative x divided by l times d -x over a.

00:20 So that means that r is equal to p -0 over a, negative x divided by l times d -x, is equal to p -0 over a times negative l, e -xp, to negative x divided by l, which then moves to p -not -l over a times 1 -1 minus e to the negative 1 power.

00:56 We know that i is equal to v0 over r, which is equal to v .0 a over p .0 l times 1 minus e to negative 1 power.

01:08 So with an upper limit of x rather than l in the integration, we know that r of x is equal to p .0 l over a times 1 minus e to the negative x divided by l.

01:25 For part b we have ex is equal to p of x of j which is equal to i p not e to the negative x divided by l over a which is equal to v 0 e to negative x divided by l over l times 1 minus e to negative 1 power for part c v equals v not minus i r of x.

01:58 We know v is equal to v0 minus v0 a over p not l times 1 minus e to negative 1 times p .0 l over a times 1 minus e to the negative x divided by l, which is then equal to v .0 times e negative x divided by l, which is then equal to v0 times e negative x divided by l, minus e negative x divided by l, divided by 1 minus e to the negative 1.

02:36 For part d, graphs of resistivity, electric field, and potential form, or sorry, from x equals 0 to l, are given, and each quantity is given in terms of the indicated unit.

03:33 So with that being said, we have these three graphs here.

03:40 I'm just going to draw them out for you.

03:57 So over here would be resistivity, which would be p -0.

04:06 Down here would be electric field, which is v .0 divided by l.

04:17 And over here it would be potential, which is v .0...

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