3 Building Intuition About Bessel Functions & Cylinders\ Separation of variables in cylindrical coordinates in general result in Bessel functions of\ different types. The goal of this problem is to develop some intuition about what Bessel\ functions look like and which ones could be solutions to a problem based on geometry. In\ Jackson's notation, the Bessel functions of the first and second kind are

J_(v)(x)

and

N_(v)(x)

\ and the modified Bessel functions are

I_()\ u (x)

and

K_()\ u (x)

. In the questions below, we'll call\ all of these Bessel functions.\ Rewrite Jackson Eqns. (3.71) to (3.75) in Griffiths notation (

\\\\Phi =>V

and

\\\ ho =>s

) .\ If

k=0

, then what is the general solution for

Z(z)

.\ If

k!=0

and purely real, then what is the general solution for

Z(z)

.\ If

k=i\\\\kappa !=0

and purely imaginary, then what is the general solution for

Z(z)

.\ If

\ u =0

, what is the general solution for

Q(\\\\phi )

?\ If

\ u

is real, what is the general solution for

Q(\\\\phi )

?\ Can

\ u

be imaginary for physical problems? Why or why not?\ If

k=\ u =0

, then what is the general solution for

R(s)

?\ If

k=0

but

\ u !=0

, then what is the general solution for

R(s)

?\ In Griffiths notation, what is

x

assuming that

k

is the separation constant?\ Can

x

be negative for physical problems? Why or why not?\ For

\ u =0,1,2

, which Bessel functions are zero for

x=0

?\ For

\ u =0,1,2

, which Bessel functions have zeros for

x>0

?\ For

\ u =0,1,2

, which Bessel functions monotonically increase as

x

increases?\ For

\ u =0,1,2

, which Bessel functions monotonically decrease as

x

increases?\ Armed with this information, we can now make educated guesses for what solu-\ tions to physical problems look like. It is most natural to consider cylinder with its\ symmetry axis lying along the

z

-axis. If the cylinder is azimuthally symmetric, then\

\ u =0

and

Q(\\\\phi )

is a constant. Now imagine a finite length cylinder with the top\ and bottom surfaces held at zero potential. The curved side of the cylinder is held\ at constant nonzero potential. Make a sketch of this configuration. What does the\ general solution to this problem look like? In other words, what are the possible\ values of

\ u

and

k

, what are the forms of

Q(\\\\phi )

and

Z(z)

, and which Bessel functions,\ if any, are involved for

R(s)

?\ A finite length azimuthally symmetric cylinder with the top and bottom surfaces\ held at a constant nonzero potential. The curved side of the cylinder is held at zero\ potential. Make a sketch of this configuration. What does the general solution to\ this problem look like? In other words, what are the possible values of

\ u

and

k

, what\ are the forms of

Q(\\\\phi )

and

Z(z)

, and which Bessel functions, if any, are involved for\

R(s)

?\ Challenge: an infinite length cylinder has a potential on the curved surface that is\ independent of

z

. What is the most general solution to this problem?

Separation of variables in cylindrical coordinates in general result in Bessel functions of different types. The goal of this problem is to develop some intuition about what Bessel functions look like and which ones could be solutions to a problem based on geometry. In Jackson's notation, the Bessel functions of the first and second kind are JV(x) and NV(x) and the modified Bessel functions are I(x) and K(x). In the questions below, we'll call all of these Bessel functions. 1. Rewrite Jackson Eqns. (3.71) to (3.75) in Griffiths notation ( $V and s ). 2. If k=0, then what is the general solution for Z(z). 3. If k=0 and purely real, then what is the general solution for Z(z). 4. If k=i=0 and purely imaginary, then what is the general solution for Z(z). 5. If =0, what is the general solution for Q() ? 6. If is real, what is the general solution for Q() ? 7. Can be imaginary for physical problems? Why or why not? 8. If k==0, then what is the general solution for R(s) ? 9. If k=0 but =0, then what is the general solution for R(s) ? 10. In Griffiths notation, what is x assuming that k is the separation constant? 11. Can x be negative for physical problems? Why or why not? 12. For =0,1,2, which Bessel functions are zero for x=0 ? 13. For =0,1,2, which Bessel functions have zeros for x>0 ? 14. For =0,1,2, which Bessel functions monotonically increase as x increases? 15. For =0,1,2, which Bessel functions monotonically decrease as x increases? 16. Armed with this information, we can now make educated guesses for what solutions to physical problems look like. It is most natural to consider cylinder with its symmetry axis lying along the z-axis. If the cylinder is azimuthally symmetric, then =0 and Q() is a constant. Now imagine a finite length cylinder with the top and bottom surfaces held at zero potential. The curved side of the cylinder is held at constant nonzero potential. Make a sketch of this configuration. What does the general solution to this problem look like? In other words, what are the possible values of and k, what are the forms of Q() and Z(z), and which Bessel functions, if any, are involved for R(s) ? 5 17. A finite length azimuthally symmetric cylinder with the top and bottom surfaces held at a constant nonzero potential. The curved side of the cylinder is held at zero potential. Make a sketch of this configuration. What does the general solution to this problem look like? In other words, what are the possible values of and k, what are the forms of Q() and Z(z), and which Bessel functions, if any, are involved for R(s) ? 18. Challenge: an infinite length cylinder has a potential on the curved surface that is independent of z. What is the most general solution to this