Consider magnetohydrostatic equilibria Vp = j x B where B and p depend only on the...
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Consider magnetohydrostatic equilibria Vp = j x B where B and p depend only on the (planar) coordinates r and y. The magnetic field is of the form B = V¥ x 2 + B¿2. (a) Compute uoj = V x B andjx B. Explain why the z-component of j x B must vanish. Conclude that contours of Bz coincide with contours of V. This means that B, like p, can be understood as a (unique) function of V. (b) Let prot = p+ B/2µo and derive the Grad-Shafranov equation: %3D dptot = -Ho B; 2µo VV = 0 or vv = AP (c) Suppose all variables (V, Bz, p) depend only on r, and that V is given. Determine Prot (x). Show that |B|2 = y? + B? is constant if p 0. %3D V? + y?. For a given V find prot (r). (d) Suppose all variables depend only on r = Assuming V (e) Show that functions of the form Vcos = r" cos no and sin = r" sin no are solutions of the force-free planar case VV = 0. Plot the magnetic field lines of y for n Vo (r/R)" and p = 0, determine B:(r). %3D %3D -2,-1, 1,2. %3D Consider magnetohydrostatic equilibria Vp = j x B where B and p depend only on the (planar) coordinates r and y. The magnetic field is of the form B = V¥ x 2 + B¿2. (a) Compute uoj = V x B andjx B. Explain why the z-component of j x B must vanish. Conclude that contours of Bz coincide with contours of V. This means that B, like p, can be understood as a (unique) function of V. (b) Let prot = p+ B/2µo and derive the Grad-Shafranov equation: %3D dptot = -Ho B; 2µo VV = 0 or vv = AP (c) Suppose all variables (V, Bz, p) depend only on r, and that V is given. Determine Prot (x). Show that |B|2 = y? + B? is constant if p 0. %3D V? + y?. For a given V find prot (r). (d) Suppose all variables depend only on r = Assuming V (e) Show that functions of the form Vcos = r" cos no and sin = r" sin no are solutions of the force-free planar case VV = 0. Plot the magnetic field lines of y for n Vo (r/R)" and p = 0, determine B:(r). %3D %3D -2,-1, 1,2. %3D
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