The 3-D Helmholtz equation for the scalar potential V inside a coaxial waveguide
ID: 3184780 • Letter: T
Question
The 3-D Helmholtz equation for the scalar potential V inside a coaxial waveguide having an inner conductor of radius 'a' and outer conductor of radius 'b' is given by:
(Delta)2V + k2V = 0
where k is a constant related to the wave number of the air inside the coaxial waveguide.
If the potential on the inner surface of the outer conductor (radius b) and the outer surface of the inner conductor (radius a) is zero, find the eigen values and the eigen functions of V. (For eigen values, find an equation that its solutions are the eigen values)
Explanation / Answer
The 3-D Helmholtz equation for the scalar potential V inside a coaxial waveguide having an inner conductor of radius 'a' and outer conductor of radius 'b' is given by (delta)2V+k2V=0. Where k is a constant related to a wave number of air inside the coaxial cable.
Coaxial cable is a wire used to carry information for the communication purpose.
First, find out the general solution for the given differential equation to find out the eigen values and eigen vectors.
Take the given differential equation
d2V+k2V=0.(homogeneous equation,del=d).
We can simplify the above equation as
(D2+k2)V=0. Where D is differential operator.
We can write the above equation as
D2+k2=0.
D2=-k2.
D=+jk,-jk.
General solution for the above differential equation will be.I am taking radius as x.
V(x)=c1coskx+c2sinkx.
Here,given initial conditions are
Potential of the inner surface of the outer conductor equal to zero,that means V(b)=0.
Outer surface of the inner conductor equal to zero,that means V(a)=0.
Now,substitute above initial conditions to find out the coefficient and eigen values.
V(a)=c1coska+c2sinka=0.
V(b)=c1coskb+c2sinkb=0.
From the above condiions,coefficient c1 definitely equal to zero for giving zero potential.
Assumet that a=b,finally the eigen values of this equation =k=n/a (or)n/b. Where n=0,1,2,3,......
Eigen functions of the given differential equation is.V(x)=sin(n/a)=sin(n/b).
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.