Given the two diff. eq's : $$\dfrac{1}{X}\dfrac{d^2X}{dx^2} = C_1 \qquad \dfrac{1}{Y}\dfrac{d^2Y}{dy^2} = C_2$$ where $C_1+C_2 = 0$. Given the boundary conditions in the figure, which coordinate should be assigned to the negative constant (and thus the sinusoidal solutions)? <img src="./images/cq_cartesian_bc_1.png" align="right" style="width: 400px";/> 1. x 2. y 3. $C_1 = C_2 = 0$ here 4. It doesn't matter. Note: * CORRECT ANSWER: B

### Exact Solutions: $$V(x,y) = \sum_{n=1}^{\infty} \dfrac{4V_0}{n\pi}\dfrac{1}{\cosh\left(\frac{n\pi}{2}\right)}\cosh\left(\frac{n\pi x}{a}\right)\sin\left(\frac{n \pi y}{a}\right)$$ ### Approximate Solutions: ### (1 term; 20 terms) <img src="./images/saddle_potential.png" align="center" style="width: 400px";/> <img src="./images/saddle_potential_20.png" align="center" style="width: 400px";/>

### Separation of Variables (Spherical) <img src="./images/metal_in_ext_field.jpg" align="center" style="width: 500px";/>

$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$ V everywhere on a spherical shell is a given constant, i.e. $V(R,\theta) = V_0$. There are no charges inside the sphere. Which terms do you expect to appear when finding V(inside)? 1. Many $A_l$ terms (but no $B_l$'s) 2. Many $B_l$ terms (but no $A_l$'s) 3. Just $A_0$ 4. Just $B_0$ 5. Something else! Note: * CORRECT ANSWER: C