Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 201 05 (c)

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MATH307 April 2013

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Question Section 201 05 (c)
Consider the following graph, interpreted as a resistor network with all resistances R = 1. (c) Suppose that by attaching a battery, the voltage at vertex 1 is held at b₁ and the voltage at vertex 2 is held at b₂. Write down vectors v and J so that the equation Lv = J describes this situation. Explain what each entry of v and J represents, and how you can use the entries to compute the effective resistance between vertices 1 and 2.

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Hint
Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 201 05 (c)/Hint 1

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Solution
${\vec {v}}={\begin{bmatrix}b_{1}\\b_{2}\\v_{3}\\v_{4}\\v_{5}\\v_{6}\end{bmatrix}}$ Here every entry is the voltage at the corresponding node. b₁ and b₂ are the voltages held by the battery connected at nodes 1 and 2. ${\vec {J}}={\begin{bmatrix}c\\-c\\0\\0\\0\\0\end{bmatrix}}$ Here every entry is the current flowing into/out of the corresponding node. c is the current coming out of node 1 from the battery, -c is the current coming out of node 2 from the battery. To find R_eff remember Ohm's law: V=IR. Hence $R_{\mathrm {eff} }={\frac {b_{2}-b_{1}}{c}}$ Notice that c is determined by the resistance of the rest of the circuit. If c is unknown, the effective resistance can be computed using Schur’s Complement. Remember: $S={\frac {1}{R_{\mathrm {eff} }}}{\begin{bmatrix}1&-1\\-1&1\end{bmatrix}}$