A long conducting duct has rectangular cross-section of width (from to ) and height…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Conductors, capacitance and boundary-value electrostatics

The Boundary-Value Problem, Ch Separation of variables in Cartesian geometry

Problem

A long conducting duct has rectangular cross-section of width

a

(from

x=0

x=a

) and height

b

(from

y=0

y=b

), with its length along

z

. Three walls,

x=0

y=0

and

y=b

, are earthed. Thin insulating strips separate them from the fourth wall,

x=a

, held at potential

V_0

. The duct is long enough that the potential inside depends only on

x

and

y

, and there is no free charge inside. ((a)) Substitute

V(x,y)=X(x)Y(y)

into Laplace's equation

\partial^2V/\partial x^2+\partial^2V/\partial y^2=0

and show that

X''/X

and

Y''/Y

must each be a constant, the two constants equal and opposite. ((b)) Two boundary conditions,

Y(0)=0

and

Y(b)=0

, apply on the pair of earthed walls that face each other. By trying both signs for the constant multiplying

Y''/Y

and checking which one admits a solution

Y

that is not zero everywhere while satisfying both conditions, show that the sign must be such that

Y''=-k^2Y

for some real

k

, and find the allowed values of

k

. ((c)) Write the function

X(x)

that solves the companion equation and vanishes on the earthed wall

x=0

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