A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by n from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by J → = J 0 k ^ , where J 0 is a constant and k ^ is along the axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius R 1 and current density J → onto the field of a solid cylindrical conductor with radius R 2 and current density − J → . Then use the fact that the appropriate azimuthal unit vectors can be expressed as θ ^ 1 = k ^ × r ^ 1 and θ ^ 2 = k ^ × r ^ 2 to show that everywhere inside the cavity the magnetic field is given by the constant B = 1 2 μ 0 J 0 k × a , where a = r 1 − r 2 and r 1 = r 1 r ^ 1 is the position of P relative to the center of the conductor and r 2 = r 2 r ^ 2 is the position of P relative to the center of the cavity.
A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by n from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by J → = J 0 k ^ , where J 0 is a constant and k ^ is along the axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius R 1 and current density J → onto the field of a solid cylindrical conductor with radius R 2 and current density − J → . Then use the fact that the appropriate azimuthal unit vectors can be expressed as θ ^ 1 = k ^ × r ^ 1 and θ ^ 2 = k ^ × r ^ 2 to show that everywhere inside the cavity the magnetic field is given by the constant B = 1 2 μ 0 J 0 k × a , where a = r 1 − r 2 and r 1 = r 1 r ^ 1 is the position of P relative to the center of the conductor and r 2 = r 2 r ^ 2 is the position of P relative to the center of the cavity.
A long, straight, cylindrical conductor contains a cylindrical cavity whose axis is displaced by n from the axis of the conductor, as shown in the accompanying figure. The current density in the conductor is given by
J
→
=
J
0
k
^
, where
J
0
is a constant and
k
^
is along the
axis of the conductor. Calculate the magnetic field at an arbitrary point P in the cavity by superimposing the field of a solid cylindrical conductor with radius R1and current density
J
→
onto the field of a solid cylindrical conductor with radius R2and current density
−
J
→
. Then use the fact that the appropriate azimuthal unit vectors can be expressed as
θ
^
1
=
k
^
×
r
^
1
and
θ
^
2
=
k
^
×
r
^
2
to show that everywhere inside the cavity the magnetic field is given by the constant
B
=
1
2
μ
0
J
0
k
×
a
, where
a
=
r
1
−
r
2
and
r
1
=
r
1
r
^
1
is the position of P relative to the center of the conductor and
r
2
=
r
2
r
^
2
is the position of P relative to the center of the cavity.