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Chapter 6: Problem 7

A chain of resistors Consider a long chain of resistors wired up like this: All the resistors have the same resistance \(R\). The power rail at the top is at voltage \(V_{+}=\) \(5 V\). The problem is to find the voltages \(V_{1} \ldots V_{N}\) at the internal points in the circuit. a) Using Ohm's law and the Kirchhoff current law, which says that the total net current flow out of (or into) any junction in a circuit must be zero, show that the voltages \(V_{1} \ldots V_{N}\) satisfy the equations $$ \begin{aligned} 3 V_{1}-V_{2}-V_{3} &=V_{+,} \\ -V_{1}+4 V_{2}-V_{3}-V_{4} &=V_{+,} \\ \vdots & \\ -V_{i-2}-V_{i-1}+4 V_{i}-V_{i+1}-V_{i+2} &=0, \\ \vdots V_{N-3}-V_{N-2}+4 V_{N-1}-V_{N} &=0, \\ -V_{N-2}-V_{N-1}+3 V_{N} &=0 . \end{aligned} $$ Express these equations in vector form \(\mathbf{A v}=\mathbf{w}\) and find the values of the matrix \(\mathrm{A}\) and the vector \(\mathrm{w}\). b) Write a program to solve for the values of the \(V_{i}\) when there are \(N=6\) internal junctions with unknown voltages. (Hint: All the values of \(V_{i}\) should lie between zero and 5V. If they don't, something is wrong.) c) Now repeat your calculation for the case where there are \(N=10000\) internal junctions. This part is not possible using standard tools like the solve function. You need to make use of the fact that the matrix \(\mathrm{A}\) is banded and use the banded function from the file banded.py, discussed in Appendix E.

Short Answer

Expert verified
Voltages satisfy \( \mathbf{A v} = \mathbf{w} \). Solve using numerical methods for large systems (use `banded.py` for large N).

Step by step solution

01

Understand the Kirchhoff Current Law

Kirchhoff's Current Law (KCL) states that the total current entering a junction must equal the total current leaving the junction. This law will help set up the equations for the voltages at various points in the resistor chain.
02

Apply Kirchhoff's Law at Each Junction

Write the KCL equations at each internal junction. For example, at junction 1 (between resistors): \[ 3V_{1} - V_{2} - V_{3} = V_{+} \] For the next junction: \[ -V_{1} + 4V_{2} - V_{3} - V_{4} = V_{+} \] Continue this process for all internal junctions.
03

Write the General Equation for Internal Junctions

For a general internal junction 'i', the KCL equation is \[ -V_{i-2} - V_{i-1} + 4V_{i} - V_{i+1} - V_{i+2} = 0 \] Ensure to handle the boundary conditions as well: At the first internal node: \[ 3V_{1} - V_{2} - V_{3} = V_{+} \] And at the last internal node: \[ -V_{N-2} - V_{N-1} + 3V_{N} = 0 \]
04

Express Equations in Matrix Form

Express the system of KCL equations in the matrix form \( \mathbf{A v} = \mathbf{w} \), where \( \mathbf{A} \) is the coefficient matrix and \( \mathbf{w} \) is the voltage vector. Coefficient matrix \( \mathbf{A} \) will be a banded matrix.
05

Define Matrix A and Vector w

For the given problem: \[ \mathbf{A} = \begin{bmatrix} 3 & -1 & -1 & 0 & 0 & \cdots & 0 \ -1 & 4 & -1 & -1 & 0 & \cdots & 0 \ -1 & -1 & 4 & -1 & -1 & \cdots & 0 \ \vdots & -1 & -1 & 4 & -1 & \vdots \ 0 & \cdots & -1 & -1 & 3 \end{bmatrix} \] and vector \( \mathbf{w} \) will be: \[ \mathbf{w} = \begin{bmatrix} V_{+} \ V_{+} \ \vdots \ 0 \end{bmatrix} \]
06

Solve for N=6

Write a program in Python (or any other language of your choice) to solve the banded matrix \( \mathbf{A} \) for \( V_{i} \) when \( N=6 \). Use `numpy.linalg.solve` if using Python.
07

Expand for N=10000

For a large number of internal junctions (like \( N=10000 \)), you will need to use specialized banded matrix solvers due to memory and computational constraints. Use the `banded` function from `banded.py` to efficiently solve the system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) is a fundamental principle in electrical circuits. It asserts that the total current entering a junction is equal to the total current leaving the junction. This means that all the current flowing into a node must flow out. To apply this, consider each internal junction in our resistor chain. By writing down KCL for each node, we ensure that we account for current conservation in the circuit. This forms the basis of our equations for determining unknown voltages in a chain of resistors. For instance, at junction 1, we set up the equation:

3V_{1} - V_{2} - V_{3} = V_{+}

Similarly, KCL is used to derive equations for other junctions, thus enabling us to solve for the internal voltages. Understanding KCL is crucial because it ensures the integrity and continuity of electrical flow in the circuit.
Banded Matrix
A banded matrix is one in which non-zero elements are confined to a diagonal band, comprising the main diagonal and zero or more neighboring diagonals. This structure reduces the memory requirements and improves computational efficiency. In our chain of resistors, the coefficient matrix \mathbf{A} is a banded matrix.

The matrix \mathbf{A} is: \[ \mathbf{A} = \begin{bmatrix} 3 & -1 & -1 & 0 & 0 & \cdots & 0 \ -1 & 4 & -1 & -1 & 0 & \cdots & 0 \ -1 & -1 & 4 & -1 & -1 & \cdots & 0 \ \vdots & -1 & -1 & 4 & -1 & \vdots \ 0 & \cdots & -1 & -1 & 3 \end{bmatrix} \]

The banded nature of \mathbf{A} allows efficient storage and fast numerical solutions. Given its structure, advanced numerical methods specifically designed for banded matrices can solve the system much faster than general-purpose solvers.
Numerical Methods
Numerical methods are mathematical tools used to approximate solutions to complex problems. These methods are essential when an analytical solution is difficult or impossible to obtain. For solving large systems of linear equations like the one in this resistor chain problem, numerical methods are indispensable.

When dealing with small matrices, direct methods such as Gaussian elimination work well. However, for larger systems (e.g., N = 10000), these methods become impractical due to their high computational cost. Instead, specialized techniques for banded matrices are used. One such technique involves the use of the `banded` function from `banded.py` which leverages the sparse nature of banded matrices to solve them efficiently.

Using these numerical methods allows us to solve for the voltages even in a large resistor chain efficiently, ensuring accuracy, and resource optimization.

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Most popular questions from this chapter

Consider the degree-six polynomial $$ P(x)=924 x^{6}-2772 x^{5}+3150 x^{4}-1680 x^{3}+420 x^{2}-42 x+1 $$ There is no general formula for the roots of a polynomial of degree six, but one can find them easily enough using a computer. a) Make a plot of \(P(x)\) from \(x=0\) to \(x=1\) and by inspecting it find rough values for the six roots of the polynomial-the points at which the function is zero. b) Write a Python program to solve for the positions of all six roots to at least ten decimal places of accuracy, using Newton's method. Note that the polynomial in this example is just the sixth Legendre polynomial (mapped onto the interval from zero to one), so the calculation performed here is the same as finding the integration points for 6-point Gaussian quadrature (see Section 5.6.2), and indeed Newton's method is the method of choice for calculating Gaussian quadrature points.

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voltages \(V_{1} \ldots V_{N}\) satisfy the equations $$ \begin{aligned} 3 V_{1}-V_{2}-V_{3} &=V_{+,} \\ -V_{1}+4 V_{2}-V_{3}-V_{4} &=V_{+,} \\ \vdots & \\ -V_{i-2}-V_{i-1}+4 V_{i}-V_{i+1}-V_{i+2} &=0, \\ \vdots V_{N-3}-V_{N-2}+4 V_{N-1}-V_{N} &=0, \\ -V_{N-2}-V_{N-1}+3 V_{N} &=0 . \end{aligned} $$ Express these equations in vector form \(\mathbf{A v}=\mathbf{w}\) and find the values of the matrix \(\mathrm{A}\) and the vector \(\mathrm{w}\). b) Write a program to solve for the values of the \(V_{i}\) when there are \(N=6\) internal junctions with unknown voltages. (Hint: All the values of \(V_{i}\) should lie between zero and 5V. If they don't, something is wrong.) c) Now repeat your calculation for the case where there are \(N=10000\) internal junctions. This part is not possible using standard tools like the solve function. You need to make use of the fact that the matrix \(\mathrm{A}\) is banded and use the banded function from the file banded.py, discussed in Appendix E.

Planck's radiation law tells us that the intensity of radiation per unit area and per unit wavelength \(\lambda\) from a black body at temperature \(T\) is $$ I(\lambda)=\frac{2 \pi h c^{2} \lambda^{-5}}{\mathrm{e}^{h c / \lambda k_{g} T}-1} $$ where \(h\) is Planck's constant, \(c\) is the speed of light, and \(k_{B}\) is Boltzmann's constant. a) Show by differentiating that the wavelength \(\lambda\) at which the emitted radiation is strongest is the solution of the equation $$ 5 \mathrm{e}^{-h c / \lambda k_{B} T}+\frac{h c}{\lambda k_{B} T}-5=0 . $$ Make the substitution \(x=h c / \lambda k_{B} T\) and hence show that the wavelength of maximum radiation obeys the Wien displacement law: $$ \lambda=\frac{b}{T} $$ where the so-called Wien displacement constant is \(b=h c / k_{B} x\), and \(x\) is the solution to the nonlinear equation $$ 5 e^{-x}+x-5=0 $$ b) Write a program to solve this equation to an accuracy of \(\epsilon=10^{-6}\) using the binary search method, and hence find a value for the displacement constant. c) The displacement law is the basis for the method of optical pyrometry, a method for measuring the temperatures of objects by observing the color of the thermal radiation they emit. The method is commonly used to estimate the surface temperatures of astronomical bodies, such as the Sun. The wavelength peak in the Sun's emitted radiation falls at \(\lambda=502 \mathrm{~nm}\). From the equations above and your value of the displacement constant, estimate the surface temperature of the Sun.

Consider a square potential well of width \(w\), with walls of height \(V\) : Using Schrödinger's equation, it can be shown that the allowed energies \(E\) of a single quantum particle of mass \(m\) trapped in the well are solutions of $$ \tan \sqrt{w^{2} m E / 2 \hbar^{2}}= \begin{cases}\sqrt{(V-E) / E} & \text { for the even numbered states, } \\ -\sqrt{E /(V-E)} & \text { for the odd numbered states, }\end{cases} $$ where the states are numbered starting from 0 , with the ground state being state 0 , the first excited state being state 1 , and so forth. a) For an electron (mass \(9.1094 \times 10^{-31} \mathrm{~kg}\) ) in a well with \(V=20 \mathrm{eV}\) and \(w=1 \mathrm{~nm}\), write a Python program to plot the three quantities $$ y_{1}=\tan \sqrt{w^{2} m E / 2 \hbar^{2}}, \quad y_{2}=\sqrt{\frac{V-E}{E}}, \quad y_{3}=-\sqrt{\frac{E}{V-E}}, $$ on the same graph, as a function of \(E\) from \(E=0\) to \(E=20 \mathrm{eV}\). From your plot make approximate estimates of the energies of the first six energy levels of the particle. b) Write a second program to calculate the values of the first six energy levels in electron volts to an accuracy of \(0.001 \mathrm{eV}\) using binary search.
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