Physics 481

For the 1D Laplace problem ($\nabla^2 V = \partial^2 V/\partial x^2 = 0$), we can choose the following ansatz: 1. $k_0\,x$ 2. $k_0\,x+k_1$ 3. $k_0\,x^2+k_1\,x+k_2$ 4. Can't tell

## Announcements * Exam 1 graded * Average: 79.5 * HW 6 Problem 1 and 2 * On your exam, you will see the grade you will receive if you choose to do problem 1 and 2 * You need to collect your exam to do these problems!

## Exam 1 Distribution <img src="./images/exam1grades.png" align="center" style="width: 800px";/>

## Exam 1 (re)Distribution* <img src="./images/exam1grades_corr.png" align="center" style="width: 800px";/>

<img src="./images/cubical_lattice.png" align="center" style="width: 300px";/> If you put a positive test charge at the center of this cube of charges, could it be in stable equilibrium? 1. Yes 2. No 3. ??? Note: * CORRECT ANSWER: B

### Method of Relaxation <img src="./images/convergence_relax.png" align="center" style="width: 700px";/>

Consider a function $f(x)$ that is both continuous and continuously differentiable over some domain. Given a step size of $a$, which could be an approximate derivative of this function somewhere in that domain? $df/dx \approx$ 1. $f(x_i+a) - f(x_i)$ 2. $f(x_i) - f(x_i-a)$ 3. $\frac{f(x_i+a) - f(x_i)}{a}$ 4. $\frac{f(x_i) - f(x_i-a)}{a}$ 5. More than one of these Note: * Correct Answer: E (C and D)

If we choose to use: $$\dfrac{df}{dx} \approx \dfrac{f(x_i+a) - f(x_i)}{a}$$ Where are we computing the approximate derivative? 1. $a$ 2. $x_i$ 3. $x_i + a$ 4. Somewhere else Note: * Correct Answer: D (it's halfway between)

Taking the second derivative of $f(x)$ discretely is as simple as applying the discrete definition of the derivative, $$f''(x_i) \approx \dfrac{f'(x_i + a/2) - f'(x_i - a/2)}{a}$$ Derive the second derivative in terms of $f$.