224 lines
6.1 KiB
Python
224 lines
6.1 KiB
Python
#!/usr/bin/python3
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib import rc
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#rc('font',**{'family':'serif','serif':['Computer Modern']})
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#rc('text', usetex=True)
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from scipy.special import ellipe,ellipk
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e = 1.602176565e-19 #osnovni naboj [C]
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m_pr = 1.672621777e-27 #masa protona [kg]
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m_el = 9.10938291e-31 #masa elektrona [kg]
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c = 299792458 #hitrost svetlobe [m/s]
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def boris(x0, v0, E, B, dt, tdur, q=1, m=1):
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''' Borisov algoritem zabjega skoka za
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integracijo diferencialne enačbe
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x0 = [x,y,z] : vektor zacetne pozicije
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v0 = [vx,vy,vz] : vektor zacetne hitrosti
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E : funkcija, ki daje jakost polja E(x)
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B : funkcija, ki daje jakost polja B(x)
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dt: casovni korak '''
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duration = int(tdur/dt)
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X = np.zeros((duration,3))
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V = np.zeros((duration,3))
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x = x0
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v = v0
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u = v0 / np.sqrt(1-(np.sum(v0**2)/c**2))
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q_prime = dt * q * 0.5 / m
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# https://en.wikipedia.org/wiki/Particle-in-cell
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# https://warpx.readthedocs.io/en/latest/theory/picsar_theory.html
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for time in range(duration):
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u_min = u + q_prime*E(x)
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t = q_prime*B(x)/np.sqrt(1+np.sum(u_min**2)/c**2)
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u_prime = u_min + np.cross(u_min,t)
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u_plu = u_min + np.cross(u_prime,2*t)/(1+np.sum(t**2))
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u = u_plu + q_prime*E(x)
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v = u/np.sqrt(1+np.sum(u_min**2)/c**2)
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x = x + dt*v
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V[time,:] = v
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X[time,:] = x
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#implementacija po wikipediji - nerelativisticno
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#h = q_prime*B(x)
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#s = 2*h / (1+np.dot(h,h))
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#u = v + q_prime*E(x)
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#u_ = u + np.cross(u+np.cross(u,h),s)
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#
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#v = u_ + q_prime * E(x)
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#x = x + dt*v
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#V[time,:] = v
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#X[time,:] = x
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return X, V
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def vay(x0, v0, E, B, dt, tdur, q=1, m=1):
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''' doi: 19,1963/1.2837054 '''
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print("ni implementiran")
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return 0
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def B_zemlja(x):
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''' model zemljinega magnentnega polja aproksimiran z
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magnetnim dipolom '''
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B0 = 3.07e-5 # [T]
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Rz = 6378137 # radii zemlje [m]
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r = np.sqrt(np.sum(x**2))
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k = -B0*Rz**3/r**5
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return k*np.array([3*x[0]*x[2],3*x[1]*x[2],2*x[2]**2-x[0]**2-x[1]**2])
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def B_loop(p, a):
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''' polje magnetne zanke polmera a v xy ravnini, normiran tok 1A
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glej: Sinigoj, ELMG polje, stran 200'''
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mu = 2e-7 # mu/2/pi
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x,y,z = p
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r = np.sqrt(x**2+y**2)
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if r < 1e-15:
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Bz = 2*np.pi*1e-7*a**2/(a**2+z**2)**1.5
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return np.array([0.,0.,Bz])
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m = np.sqrt(4*a*r/( (r+a)**2+z**2))
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K = ellipk(m**2)
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E = ellipe(m**2)
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C = 1./np.sqrt((a+r)**2+z**2)
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Br = -mu * z/r*C * (-K + (a**2+r**2+z**2)*E/((a-r)**2+z**2))
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Bz = mu*C * (K - (a**2-r**2-z**2)*E/((a-r)**2+z**2))
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return np.array([Br*x/r, Br*y/r, Bz])
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def B_bottle(p, a, h, I):
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''' polje magnetne steklenice visine 2h in polmera a
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'''
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B1 = B_loop(p-np.array([0.,0.,h]),a)*I
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B2 = B_loop(p+np.array([0.,0.,h]),a)*I
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return B1+B2
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def tokamak(p, a, b, I, n):
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''' polje v modelu tokamaka, zanke polmera a
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nataknjene na toroid polmera b '''
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Rx = np.array([[1.,0.,0.],[0.,0.,-1.],[0.,1.,0.]])
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B = np.zeros(3)
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premik = np.array([b,0.,0.])
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for i in range(n):
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theta = i*2.*np.pi/n
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Rz = np.array([[np.cos(theta), np.sin(theta),0.],
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[-np.sin(theta),np.cos(theta), 0.],
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[0.,0.,1.]])
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#print(Rz)
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p_ = np.matmul(Rx.T, np.matmul(Rz.T,p)) + premik
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b_ = I*B_loop(p_,a)
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b = np.matmul(Rz, np.matmul(Rx,b_))
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B = B+b
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return B
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def Wk(m,V):
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return 0.5*m*np.sum(V**2,1)
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def plot3(X,enote='m',zemlja=False,lim=[],bottle=[],tokamak=[]):
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fig = plt.figure()
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ax = fig.add_subplot(projection='3d')
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if zemlja:
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u = np.linspace(0, 2 * np.pi, 100)
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v = np.linspace(0, np.pi, 100)
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x = np.outer(np.cos(u), np.sin(v))
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y = np.outer(np.sin(u), np.sin(v))
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z = np.outer(np.ones(np.size(u)), np.cos(v))
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ax.plot_surface(x,y,z,zorder=-4)
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#ax.set_box_aspect(np.ptp(X,axis=0))
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ax.plot(X[:,0],X[:,1],X[:,2],zorder=4)
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ax.set_xlabel(r'$x$ [{}]'.format(enote))
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ax.set_ylabel(r'$y$ [{}]'.format(enote))
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ax.set_zlabel(r'$z$ [{}]'.format(enote))
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if lim:
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ax.set_xlim(lim)
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ax.set_ylim(lim)
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ax.set_zlim(lim)
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if bottle:
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r = bottle[0]
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h = bottle[1]
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fi = np.linspace(0,2*np.pi,300)
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a = r*np.cos(fi)
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b = r*np.sin(fi)
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ax.plot(a,b,+h,color='red',zorder=100)
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ax.plot(a,b,-h,color='red',zorder=100)
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if tokamak:
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ax=plot_tokamak(tokamak[0],tokamak[1],tokamak[2],ax)
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plt.show()
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def plot_tokamak(a,b,n,ax=None):
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fi = np.linspace(0,2.*np.pi,300)
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x = a*np.cos(fi)
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y = a*np.sin(fi)
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z = np.zeros(300)
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X=np.zeros([300,3,n])
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x = x-b
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Rx = np.array([[1.,0.,0.],[0.,0.,-1.],[0.,1.,0.]])
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if ax == None:
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fig = plt.figure()
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ax = fig.add_subplot(projection='3d')
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for i in range(n):
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theta = i*2.*np.pi/n
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Rz = np.array([[np.cos(theta),-np.sin(theta),0.],
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[np.sin(theta),np.cos(theta),0.],
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[0.,0.,1]])
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for j in range(300):
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p = np.array([x[j],y[j],z[j]])
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#X=np.zeros([300,3])
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X[j,:,i] = np.matmul(Rz, np.matmul(Rx,p.T))
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ax.plot(X[:,0,i],X[:,1,i],X[:,2,i],color='red',zorder=1)
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plt.draw()
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plt.show()
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def rplot(X, enote='m', xy=[0,1],os='xy', fname=None):
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fig = plt.figure()
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ax = fig.subplots()
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ax.plot(X[:,xy[0]],X[:,xy[1]], 'k')
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ax.grid()
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dx = X[-1,xy[0]]-X[-2,xy[0]]
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dy = X[-1,xy[1]]-X[-2,xy[1]]
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#ax.arrow(X[-1,xy[0]],X[-1,xy[1]],dx,dy,head_width=0.1,head_length=0.2,
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# overhang=0.3, zorder=10,color='k')
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ax.set_xlabel(r'${}$ [{}]'.format(os[0],enote))
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ax.set_ylabel(r'${}$ [{}]'.format(os[1],enote))
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if fname:
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plt.savefig(fname,bbox_inches='tight')
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plt.show()
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if __name__ == "__main__": # testni del kode
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E = lambda x: np.array([0.0,0.2,0.3])
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B = lambda x: np.array([0.,0.0,1.*x[0]])
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x0 = np.array([-1.,0.,0.])
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v0 = np.array([0.,1.,0.])
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dt = 1e-2
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#X, V = boris(x0,v0,E,B,dt,20)
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#plot3(X)
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plot_tokamak(1,2,16)
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