A steady current `I` flows down a long cylindrical wire of radius `a`. Find the magnetic field, both inside and outside the wire
Problem:
A steady current `I` flows down a long cylindrical wire of radius `a`. Find the magnetic field, both inside and outside the wire, if
(a) The current is uniformly distributed over the outside surface of the wire.
(b) The current is distributed in such a way that `J` is proportional to `s` , the distance from the axis.
Answer:
a) The current is uniformly distributed over the outside of the surfaces of the wire:
Inside the cylinder `(s<a)`:
The current inside the cylinder is
`I=0`
So, The magnetic field inside the cylinder is
`ointvecB_(s<a).vecdl=mu_0I`
so, `B_(s<a)=0`
Outside the cylinder `(s>a)`:
For this case, the current enclose the ampere's loop is `I`
So, the magnetic field outside the cylinder is
`ointvecB_(s>a).vecdl=mu_0I`
or, `B_(s>a).2pis=mu_0I`
or, `B_(s>a)=(mu_0I)/(2pis)`
b) The current is distributed in such a way that `J` is proportional to `s`:
The current density is
`J props`
or, `J=cs`
Therefore we can say that,
`J={(cs,"for " s<a),(0,"for " s>a):}`
The total current
`I=int_0^avecJ.vecds`
or, `I=int_(r=0)^a int_(phi=0)^(2pi)csdssdphi`
or, `I=c[s^3/3]_0^a[phi]_0^(2pi)`
or, `I=2/3cpia^3`
or, `c=(3I)/(2pia^3)`
Inside the cylinder `(s<a)`:
The current inside the cylinder is
`I_"in"=int_0^svecJ.vecds`
or, `I_"in"=int_(r=0)^s int_(phi=0)^(2pi)csdssdphi`
or, `I_"in"=c[s^3/3]_0^s[phi]_0^(2pi)`
or, `I_"in"=2/3cpis^3`
or, `I_"in"=2/3(3I)/(2pia^3)pis^3`
or, `I_"in"=Is^3/a^3`
The magnetic field inside the cylinder is
`ointvecB_(s<a).vecdl=mu_0I_"in"`
or, `B_(s<a).2pis=mu_0(Is^3/a^3)`
or, `B_(s<a)=(mu_0Is^2)/(2pia^3)`
Outside the cylinder `(s>a)`:
For this case, the current enclose the ampere's loop is `I`
So, the magnetic field outside the cylinder is
`ointvecB_(s>a).vecdl=mu_0I`
or, `B_(s>a).2pis=mu_0I`
or, `B_(s>a)=(mu_0I)/(2pis)`