Potential of Hollow Cube

A hollow cube has conducting walls defined by 6 planes $x=y=z=0$ and $x=y=z=a$. The walls at $z=0$ and $z=a$ are held at a constant potentail $V_0$. The other four sides are at zero potential. Find the potentail $V(x,y,z)$ at any point inside the cube.

Inside the cube (no charge) Laplace equation for potential is

$∇^2 V=0 or (d^2 V)/(dx^2 )+(d^2 V)/(dy^2 )+(d^2 V)/(dz^2 )=0$

$V(x,y,z)=X(x)*Y(y)*Z(z) ⇒$

$YZ*(d^2 X)/(dx^2 )+XZ*(d^2 Y)/(dy^2 )+XY*(d^2 Z)/(dz^2 )=0$

$1/X*(d^2 X)/(dx^2 )+1/Y*(d^2 Y)/(dy^2 )+1/Z*(d^2 Z)/(dz^2 )=0$ or

$α^2+β^2-γ^2=0$ with $α$,$β$ and $γ$ numbers

$((d^2 X)/(dx^2 )=α^2 X$

$(d^2 Y)/(dy^2 )=β^2 X$

$(d^2 Z)/(dz^2 )=-γ^2 X$


$(X(x)=A*\cos⁡ αx+B*\sin ⁡αx$

$Y(y)=C*\cos⁡ βy+D*\sin ⁡βy$

$Z(z)=E*\cosh⁡ γz+F*\sinh⁡ γz$

Boundary conditions

$ϕ(0,y,z)=0$ and $ϕ(a,y,z)=0$ mean $A=0$ and $α=nπ/a$

$ϕ(x,0,z)=0$ and $ϕ(x,a,z)=0$ mean $C=0$ and $β=nπ/a$

$γ=π/a*\sqrt{(m^2+n^2 )}$


$ϕ(x,y,z)=∑_{(m,n=1..∞)} \sin(⁡nπ/a x)*\sin ⁡(mπ/a*y)*$

$*[E_{mn} \cosh⁡ ((π\sqrt{(m^2+n^2 )})/a z)+F_{mn} \sinh⁡((π√(m^2+n^2 ))/a z)]$

Remaining two boundary conditions are

$ϕ(x,y,0)=V_0 =∑_(m,n=1..∞) E_{mn} \sin⁡ (nπ/a x)*\sin⁡(mπ/a*y)$

To find E_{mn} we use the Fourier trick

$E_{mn}=(4V_0)/a^2 *∫_0^a ∫_0^a \sin⁡〖(nπ/a x)*\sin⁡(mπ/a*y) *dx*dy=$

$=0$ ,for m or n even

$=(4V_0)/a^2 *a^2/(π^2 mn)=(4V_0)/(π^2 mn)$ for m,n odd

We have also

$ϕ(x,y,a)=V_0=ϕ(x,y,0)$ so that $E_{mn} \cosh⁡ γ_{mn}+F_{mn}*\sinh⁡ γ_{mn} =E_{mn}$

$F_{mn}=E_{mn} (1-\cosh⁡ γ_{mn} )/\sinh⁡ γ_{mn}$

where $γ_{mn}=(π\sqrt{(m^2+n^2 )})$


$ϕ(x,y,z)=(16V_0)/π^2 *∑_{(m,n odd=1)}^∞ (1/mn) \sin⁡ (nπ/a x)\sin(mπ/a*y)*$

$*[\cosh⁡((γ_{mn} z)/a)+(1-\cosh⁡ γ_{mn} )/\sinh⁡ γ_{mn} *\sinh⁡ ((γ_{mn} z)/a)]$