With the approximate form of Laplace's equation: $\dfrac{V(x_i+a) - 2V(x_i) + V(x_i-a)}{a} \approx 0$ What is a the appropriate estimate of $V(x_i)$? 1. ${1}/{2}(V(x_i+a)-V(x_i-a))$ 2. ${1}/{2}(V(x_i+a)+V(x_i-a))$ 3. ${a}/{2}(V(x_i+a)-V(x_i-a))$ 4. ${a}/{2}(V(x_i+a)+V(x_i-a))$ 5. Something else Note: * Correct answer: B
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To investigate the convergence, we must compare the estimate of $V$ before and after each calculation. For our 1D relaxation code, $V$ will be a 1D array. For the kth estimate, we can compare $V_k$ against its previous value by simply taking the difference. Store this in a variable called ``err``. What is the type for ``err``? 1. A single number 2. A 1D array 3. A 2D array 4. ??? Note: * Correct Answer: B
The Method of Relaxation also works for Poisson's equation (i.e., when there is charge!). Given, $\nabla^2 V \approx \dfrac{V(x+a)-2V(x)+V(x-a)}{a^2}$ Which equations describes the appropriate "averaging" that we must do: 1. $V(x) = \dfrac{1}{2}(V(x+a)-V(x-a))$ 2. $V(x) = \dfrac{\rho(x)}{\varepsilon_0}+\dfrac{1}{2}(V(x+a)+V(x-a))$ 3. $V(x) = \dfrac{a^2\rho(x)}{2\varepsilon_0}+\dfrac{1}{2}(V(x+a)+V(x-a))$ Note: * Correct answer: C
### Separation of Variables (Cartesian) <img src="./images/cartesian_sep_variables.png" align="center" style="width: 500px";/>
Say you have three functions $f(x)$, $g(y)$, and $h(z)$. $f(x)$ depends on $x$ but not on $y$ or $z$. $g(y)$ depends on $y$ but not on $x$ or $z$. $h(z)$ depends on $z$ but not on $x$ or $y$. If $f(x) + g(y) + h(z) = 0$ for all $x$, $y$, $z$, then: 1. All three functions are constants (i.e. they do not depend on $x$, $y$, $z$ at all.) 2. At least one of these functions has to be zero everywhere. 3. All of these functions have to be zero everywhere. 4. All three functions have to be linear functions in $x$, $y$, or $z$ respectively (such as $f(x)=ax+b$) Note: * CORRECT ANSWER: A
If our general solution contains the function, $$X(x) = Ae^{\sqrt{c}x} + Be^{-\sqrt{c}x}$$ What does our solution look like if $c<0$; what about if $c>0$? 1. Exponential; Sinusoidal 2. Sinusoidal; Exponential 3. Both Exponential 4. Both Sinusoidal 5. ??? Note: * CORRECT ANSWER: B