Question

***PLEASE NOTE: I have already calculated the potential and need help with Electric Field and induced charge at the boundaries for this problem***

An infinitely long rectangular metal pipe (sides and ) is grounded, but one end, at x = 0, is maintained at a specified potential $V_{0}(y,z)$, as indicated in Fig. 3.22. What is the ELECTRIC FIELD and INDUCED CHARGE on all boundaries? I have already worked you the potential, and got:

Answer 1

I have one point more, in the solution, actually the normal on the surfaces at y = 0 or z = 0 are directed along negative y or negative z. Therefore, the sign of surface charge density obtained in the solution in that should be positive perhaps as well.

Given V(x, y, z) = 16V_0/pi^2 sigma^infinity_n, m = 1, 3, 5 ... 1/nm exp(-pi squareroot (n/a)^2 + (n/b)^2 x)sin(n pi y/x) sin(m pi^2/b) The electric field can be calculate as follows E_x = -partial differential V/partial differential x, E_y = -partial differential V/partial differential y, E_z = -partial differential V/partial differential z Thus, E_x = -16V_0/pi^2 sigma^infinity_n, m = 1, 3, 5 ... pi/nm squareroot (n/a)^2 + (m/b)^2 exp (-pi squareroot (n/a)^2 + (m/b)^2 x)sin(n pi y/a) sin(m pi^2/b) = -16V_0/pi sigma^infinity_n, m = 1, 3, 5 ... 1/nm squareroot (n/a)^2 + (m/b)^2 exp(-pi squareroot (n/a)^2 + (m/b)^2 x)sin(n pi y/a) sin(n pi^2/b) E_y = 16V_0/pi^2 sigma_n, m = 1, 3, 5 ... 1/nm exp(-pi squareroot (n/a)^2 + (m/b)^2 x)n pi/a cos(n pi y/a) sin(n pi^2/b) = 16V_0/pi a sigma_n, m = 1, 3, 5 ... 1/m exp(-pi squareroot (n/a)^2 + (m/b)^2 x)cos(n pi y/a) sin(m pi^2/b) Similarly E_z = -partial differential V/partial differential z