- #1
maria clara
- 58
- 0
On the Z=0 plane the charge distribution is of the form
[tex]\rho[/tex]s=[tex]\rho[/tex]0 sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )
find the potential everywhere, assuming that [tex]\phi[/tex](z[tex]\rightarrow[/tex]±[tex]\infty[/tex])=0
according to the answer, we should look for a potential of the form
A sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )f(z)
(due to the form of the charge distribution)
in order to satisfy the condition [tex]\phi[/tex](z[tex]\rightarrow[/tex]±[tex]\infty[/tex])=0
we conclude that the potential has the following form:
A sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )exp(-z) , z>0
A sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )exp(z) , z<0
this potential clearly satisfies the boudary condition, but does f(z) have to be exponential? how can we be sure it isn't some other function that decays at infinity?
and another question - in which cases is it possible to deduce that the potential has the same form as the charge distribution? is it always true when the charge distribution is infinite and is a product of separate functions of each variable?
thanks
[tex]\rho[/tex]s=[tex]\rho[/tex]0 sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )
find the potential everywhere, assuming that [tex]\phi[/tex](z[tex]\rightarrow[/tex]±[tex]\infty[/tex])=0
according to the answer, we should look for a potential of the form
A sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )f(z)
(due to the form of the charge distribution)
in order to satisfy the condition [tex]\phi[/tex](z[tex]\rightarrow[/tex]±[tex]\infty[/tex])=0
we conclude that the potential has the following form:
A sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )exp(-z) , z>0
A sin( [tex]\alpha[/tex] x )sin( [tex]\beta[/tex] y )exp(z) , z<0
this potential clearly satisfies the boudary condition, but does f(z) have to be exponential? how can we be sure it isn't some other function that decays at infinity?
and another question - in which cases is it possible to deduce that the potential has the same form as the charge distribution? is it always true when the charge distribution is infinite and is a product of separate functions of each variable?
thanks