Consider a toroidal structure with a rectangular cross-section. If the toroid is defined by
the surfaces r = 1cm and r = 4cm and the planes z = 0 cm and z = 2 cm, and the surface
current density on the surface defined by r = 4cm is given by -60 A/m in the z direction.
(a)
Specify the current densities on the surfaces at r = 1, z = 0, and z = 2.
(b)
Find the expression for H inside the toroid (i.e. in the region 1 < r < 4 cm and 0 <
z < 2cm).
I have the solution, but I'm having trouble understanding it. The current density, K, for r=1 was found to be 240 A/m.
To find the field inside the toroid, ampere's law is used with ∮H dl = 2∏r = ∫(K at r=1)dθ from θ=0 to 2∏. Giving the final answer of H=15/(2∏r)
So in this question K is different at r=4 and r=1. Is it changing due to distance from the sources of the field, or is it uniform and I'm missing some simple math ratio? Since K is different at different radii, how come we integrate only integrate with Kr=1 and not other any other K's? I mean it is changing with distance and I'm assuming since the cross section of the toroid is a square loop, all four sides would contribute to the field inside the toroid, not just K at r=1.
How come I don't need to calculate for Kr=4 and Kz=0 =Kz=2 and sum them up? Hope that made sense, thanks in advance.
the surfaces r = 1cm and r = 4cm and the planes z = 0 cm and z = 2 cm, and the surface
current density on the surface defined by r = 4cm is given by -60 A/m in the z direction.
(a)
Specify the current densities on the surfaces at r = 1, z = 0, and z = 2.
(b)
Find the expression for H inside the toroid (i.e. in the region 1 < r < 4 cm and 0 <
z < 2cm).
I have the solution, but I'm having trouble understanding it. The current density, K, for r=1 was found to be 240 A/m.
To find the field inside the toroid, ampere's law is used with ∮H dl = 2∏r = ∫(K at r=1)dθ from θ=0 to 2∏. Giving the final answer of H=15/(2∏r)
So in this question K is different at r=4 and r=1. Is it changing due to distance from the sources of the field, or is it uniform and I'm missing some simple math ratio? Since K is different at different radii, how come we integrate only integrate with Kr=1 and not other any other K's? I mean it is changing with distance and I'm assuming since the cross section of the toroid is a square loop, all four sides would contribute to the field inside the toroid, not just K at r=1.
How come I don't need to calculate for Kr=4 and Kz=0 =Kz=2 and sum them up? Hope that made sense, thanks in advance.