- #1
Dustinsfl
- 2,281
- 5
Solve Laplace's equation on a circular disk of radius [itex]a[/itex] subject to the piecewise boundary condition
$$
u(a,\theta) = \begin{cases}
1, & \frac{\pi}{2} - \epsilon < \theta < \frac{\pi}{2} + \epsilon\\
0, & \text{otherwise}
\end{cases}
$$
where [itex]\epsilon \ll 1[/itex].
Physically, this would reflect the electric potential distribution on a conducting disk whose edge is almost completely grounded except a small portion of angular extent [itex]\Delta\theta = 2\epsilon[/itex] around the location [itex]\theta = \frac{\pi}{2}[/itex].
Obtain the solution to this problem and plot the solution for the case of [itex]a = 1[/itex] and [itex]\epsilon = 0.05[/itex].
By separation of variables, we have
$$
\begin{cases}
\Theta(\theta) = A\cos\lambda\theta + B\sin\lambda\theta\\
R(r) = r^{\pm\lambda}
\end{cases}
$$
So how do I use the conditions now?
$$
u(a,\theta) = \begin{cases}
1, & \frac{\pi}{2} - \epsilon < \theta < \frac{\pi}{2} + \epsilon\\
0, & \text{otherwise}
\end{cases}
$$
where [itex]\epsilon \ll 1[/itex].
Physically, this would reflect the electric potential distribution on a conducting disk whose edge is almost completely grounded except a small portion of angular extent [itex]\Delta\theta = 2\epsilon[/itex] around the location [itex]\theta = \frac{\pi}{2}[/itex].
Obtain the solution to this problem and plot the solution for the case of [itex]a = 1[/itex] and [itex]\epsilon = 0.05[/itex].
By separation of variables, we have
$$
\begin{cases}
\Theta(\theta) = A\cos\lambda\theta + B\sin\lambda\theta\\
R(r) = r^{\pm\lambda}
\end{cases}
$$
So how do I use the conditions now?