The resistivity of a semiconductor can be modified by adding...

The resistivity of a semiconductor can be modified by adding different amounts of impuritics. 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 therod 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

Relationship Between Electric Field and Electric Potential

The electric potential within a material is related to the electric field through a spatial derivative; specifically, the electric field is the negative gradient of the electric potential. This connection is essential for transitioning between understanding the energy per unit charge (potential) and the force experienced by charges (electric field), particularly in systems with varying properties.

Integration Methods in Calculating Resistance

When resistivity varies with position, the total resistance of the material must be determined by integrating the local contributions along the length of the material. This use of integral calculus is fundamental in handling systems with spatially dependent properties and is critical for determining overall characteristics such as total resistance, current flow, and voltage drop.

Graphical Analysis of Physical Quantities

Graphical analysis involves visualizing how physical quantities like resistivity, electric field, and electric potential vary with position. This technique aids in developing an intuitive understanding of how non-uniform material properties influence the behavior of current flow and energy distribution within the system, facilitating insight into the dynamic interplay between these quantities.

Variable Resistivity

This concept involves the idea that the intrinsic resistance of a material (expressed as resistivity) can change with position. In problems like this, resistivity is not uniform but varies as a function of spatial coordinates. Understanding variable resistivity is crucial for analyzing how current distributes in a non-homogeneous medium and computing the overall resistance by incorporating its spatial dependence.

Ohm's Law in Differential Form

Ohm's law relates the current density, electric field, and conductivity (or resistivity) locally within a material. In its differential form, it states that the electric field is equal to the product of the local current density and the local resistivity. This formulation is essential when dealing with materials that have non-uniform properties, as it allows one to relate microscopic behavior to overall circuit behavior.

Transcript

00:01 Hi everyone this is the problem based on current electricity here we have to calculate the resistance electric field potential and current flowing through the conductor here it is given the resistivity of a semiconductor material is modified by adding the impurity so that its resistivity varies with distance from zero to l according to to the relation row is equal to row not exponential of minus x upon l here x is the distance from the origin it is placed along the x axis in the first part we have to find the resistance and current total resistance and current flowing through it for a given potential in p part we have to find the electric field inside it as a function of x and c we have to calculate potential across it as a function of x in d part we have to plot the graph for resistivity as a function of x and potential as a function of x let us see it if we consider the element of thickness d x at a distance x from o then its resistance we can find resistance is row l upon a so its sickness is d x upon a that is area cross section which is uniform row is given row not exponential of minus x by l d x upon a to find the total resistance we have to integrate it for the limit 0 to l so r you will get row not 0 by a 1 minus exponential of minus 1.

02:45 For applied potential difference of v, current flowing through it is v upon r, that is v0 0 upon row 0 1 1 minus exponential to the power minus 1 divided by a.

03:04 So it is to be v0 a upon ro not l 1 minus exponential to the power minus 1 so this is the current now in b part we have to find the electric field electric field is defined as resistivity into density so e as a function of x if you find it will be row i row not exponential of x upon l upon a if we substitute the value of i we will get v0 exponent of minus x upon l l upon 1 minus e to the power minus 1 c part so the distance of the wire at a distance x from origin we can write r is equal to 0 to x dr dr having the value as we have calculated in the part a that is r0 by a exponation of minus x upon l d x so on simplifying and substituting the limit we will get l roe not a 1 minus exponential of x upon n so potential difference at a distance is x you will get vx minus v0 is equal to i into r so v x you will get b not substitute the value of i we will get b not by a row not l 1 minus exponation of e to the power minus 1 into the power minus 1 into l r not a 1 and here it is minus and i have to correct.

06:24 So on solving this equation, potential as a function of x can be written as v0 into exponential of minus x upon l minus e to the power minus 1 upon 1 minus exponential of minus 1...

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