電場與庫倫定律

1. 電荷所建立的電場

我們知道電荷的存在會在空間中建立起一個電場，最簡單的電荷源就是電點荷。描述單一一個電荷所建立的電場的基本定律就是庫倫定律。一般我們所看到最簡單的庫倫定律的形式如下： \[\vec{E}=\frac{kq}{r^2}\hat{r}=\frac{kq}{r^3}\vec{r} \] 請注意在上面這個公式中我們用了一個非常重要的向量關係，那就是位置向量(\(\vec{r}\))的單位向量(\(\hat{r}\))等於位置向量處理位置向量的長度(\(|\vec{r})\))，而且還要注意到上面這個公式，以坐標系統的原點就在點電荷的位置，來描述空間中場的位置向量，如果我們在空間中有許多的電荷，那麼這個公式就不適合來描述這樣的情況。所以比較好的庫倫定律的寫法應該是下面的公式。

座標的原點不在點電荷q上面，所以點電荷q有自己的位置向量(\(\vec{r}'\))，欲計算的電場位置有向量是以\(\vec{r}\)來描述，電場本身也是一個向量(\(\vec{E}\))，其方向及大小與空間中的位置有關，位置\(\vec{r}\)的電場向量是\(\vec{r}\)的函數，當然也是x,y,z的函數： \[\vec{E}(\vec{r})=\frac{kq}{|\vec{r}-\vec{r}'|^3}(\vec{r}-\vec{r}') \] 如果我們有很多的電荷那麼就把這個公式推廣一下，就得到多個點電荷在空間在空間中所建立的電場： \[\vec{E}(\vec{r})=\sum_i \frac{kq}{|\vec{r}-\vec{r_i}'|^3}(\vec{r}-\vec{r_i}') \] 在我們下面的程式中就是反覆的應用這個電場公式來計算單一一個點電荷，或是有多個電荷分佈成不同的幾何形狀時，在空間中分別建立起不同的電場。在普通物理中我們已經學過，單一點電荷所建立的電場大小，隨著與電荷距離的變化是平方反比定律；但是如果把電荷均勻地排成一條無窮長的直線時，電場的大小與距離成1次方反比定律；如果把電荷平均地分佈在一個平面上，這個平面將會造就出一個常數的電場，電場大小與距離完全無關。如果我們有一個正電荷和一個負電荷分隔一段距離形成一個所謂的電偶極，那麼在遠離這個電偶極的地方去測量電場的話，會發現電場的大小隨著遠離電偶極的距離是3次方反比定律。我們希望能夠藉由下面的數值計算來驗證這些重要結果。

\(\rm{point \, charge:\,\,}E \propto 1/r^2\)
\(\rm{line \, of \, charges:\,\,}E \propto 1/r;\,\, E=\frac{2k\lambda}{r}\)
\(\rm{point \, plane \, of \, charges:\,\,}E \simeq \, constant \,\, E=\frac{\sigma}{2 \epsilon_0}\)
\(\rm{2 point \, charges(dipole):\,\,}E \propto 1/r^3\)

點電荷所建立的電場

GlowScript 3.0 VPython
def EF_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    E=ke*q*rrq/rrq0**3
    return E,Es
q=1.; rq=vec(0.,0.,0.)
r1=vec(1,0,0)
E1,E1s=EF_point(q,rq,r1)
print(r1,'E1=',E1,'  E1s=',E1s)
r2=vec(1,0,1)
E2,E2s=EF_point(q,rq,r2)
print(r2,E2,E2s)
r3=vec(1,1,1)
E3,E3s=EF_point(q,rq,r3)
print(r3,E3,E3s)
scene=canvas(width=800, height=600, center=vector(1,1,0))
box(pos=vec(1,0,1),size=vec(2,0.01,3),color=vec(0,1,1),opacity=0.5)
X=arrow(pos=vec(0,0,0),axis=vec(2,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,2),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Q1=sphere(pos=rq,radius=0.3,color=vec(1,1,0))
F1=sphere(pos=r1,radius=0.1,color=vec(1,1,1))
F2=sphere(pos=r2,radius=0.1,color=vec(1,1,1))
F3=sphere(pos=r3,radius=0.1,color=vec(1,1,1))
A1=arrow(pos=r1,axis=E1,shaftwidth=0.02,headwidth=0.04,color=vec(1,0,1))
A2=arrow(pos=r2,axis=E2,shaftwidth=0.02,headwidth=0.04,color=vec(1,0,1))
A3=arrow(pos=r3,axis=E3,shaftwidth=0.02,headwidth=0.04,color=vec(1,0,1))

run this code(EL-01) in glowscript

點電荷的電場畫圖（graph)

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V
q=1.; rq=vec(0.,0.,0.)
N=20; dx=0.1
gd = graph(width=800, height=600, background=color.black,
      title='Electric field and potential of a point charge',
      xtitle='x', ytitle='E/V' ,xmin=0, xmax=2.2, ymin=0, ymax=10)
f1 = gcurve(color=color.cyan, label="E(x)") 
f2 = gdots(color=color.green, label="V(x)") 
for i in range(N):
    x=dx*(i+1)
    r=vec(x,0,0)
    E,Es,V=EFV_point(q,rq,r)
    f1.plot(x,Es)
    f2.plot(x,V)

run this code(EL-02) in glowscript

2點電荷(+,- or +,+)的電場

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V
q1=1.; rq1=vec(0.,0.,0.)
q2=1.; rq2=vec(4.,0.,0.)
N=38; dx=0.1
scene=canvas(width=800, height=600, center=vector(2.5,1,0))
X=arrow(pos=vec(0,0,0),axis=vec(5,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,0.5),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Q1=sphere(pos=rq1,radius=0.1,color=vec(1,1,0))
Q2=sphere(pos=rq2,radius=0.1,color=vec(1,1,0))
s=0.1
gd = graph(width=800, height=600, background=color.black,
      title='Electric field 2 point charges at x=0 and x=4',
      xtitle='x', ytitle='E/V' ,xmin=0, xmax=4.2, ymin=0)
f1 = gcurve(color=color.cyan, label="y=0.5") 
f2 = gcurve(color=color.green, label="y=1.0") 
for i in range(N):
    x=dx*(i+1)
    r1=vec(x,0.5,0)
    E1,Es1,V1=EFV_point(q1,rq1,r1)
    E2,Es2,V2=EFV_point(q2,rq2,r1)
    E=E1+E2
    Es=mag(E)
    f1.plot(x,Es)
    arrow(pos=r1,axis=E*s,shaftwidth=0.02,headwidth=0.04,color=vec(1,0,1))
    r2=vec(x,1.0,0)
    E1,Es1,V1=EFV_point(q1,rq1,r2)
    E2,Es2,V2=EFV_point(q2,rq2,r2)
    E=E1+E2
    Es=mag(E)
    f2.plot(x,Es)
    arrow(pos=r2,axis=E*s,shaftwidth=0.02,headwidth=0.04,color=vec(1,0,1))

run this code(EL-03) in glowscript

比較點電荷與電偶極的電場

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V
q0=1.; rq0=vec(0.,0.,0.)
q1=1.; rq1=vec(-1,0.,0.)
q2=-1; rq2=vec(1,0.,0.)
N=200; dx=0.1
scene=canvas(width=800, height=600, center=vector(0,3,0))
X=arrow(pos=vec(-1.2,0,0),axis=vec(2.4,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Y=arrow(pos=vec(0,0,0),axis=vec(0,8,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,1))
Q0=sphere(pos=rq0,radius=0.1,color=vec(1,1,0))
Q1=sphere(pos=rq1,radius=0.1,color=vec(1,0,0))
Q2=sphere(pos=rq2,radius=0.1,color=vec(0,0,1))
gd = graph(width=800, height=600, background=color.black,
      title='Electric field 2 point charges at x=0 and x=4',
      xtitle='x', ytitle='E', ymax=0.1) # ,xmin=0, xmax=4.2, ymin=0)
f1 = gcurve(color=color.cyan, label="only 1 q") 
f2 = gcurve(color=color.green, label="q and -q") 
for i in range(N):
    x=0.1+dx*(i)
    r=vec(0,x,0)
    E,Es,V=EFV_point(q0,rq0,r)
    Es1=mag(E)
    f1.plot(x,Es1)
    E1,Es1,V1=EFV_point(q1,rq1,r)
    E2,Es2,V2=EFV_point(q2,rq2,r)
    Ed=E1+E2
    Es2=mag(Ed)
    f2.plot(x,Es2)

run this code(EL-03B) in glowscript

2. 電力線

拋體運動的軌跡以向量表示

\(x=v\cos(\theta)t\)
\(y=v\sin(\theta)t-\dfrac{1}{2} g t^2\)
\(\tan(\theta)=\dfrac{y(0)}{x(0)}\)
\(t=\dfrac{x}{v \cos(\theta)}\)
\(y(x)=\tan(\theta)x-\dfrac{g}{2 v^2 \cos^2(\theta)} x^2\)
\(m=\dfrac{dy}{dx}=\tan(\theta)-\dfrac{g}{ v^2 \cos^2(\theta)} x\)
\(s^2=dx^2 + dy^2; \,\,\, dy=m dx\)
\(dx=\sqrt{\dfrac{s^2}{1+m^2}}; \,\,\, dy=mdx\)
\(\vec{r}'=\vec{r}+d\vec{r}; \,\,\, d\vec{r}=\rm{vec(dx,dy,0)}\)

GlowScript 3.0 VPython
def line(v,Ni,s,r1):
    r=1*r1; g=9.8; theta=atan(r.y/r.x)
    for i in range(Ni):
        m=tan(theta)-(g/(v*cos(theta))**2)*r.x
        dx=sqrt(s**2/(1+m**2))
        dy=dx*m
        dr=vec(dx,dy,0)
        arrow(pos=r,axis=dr,shaftwidth=0.02,headlength=0.12,headwidth=0.12,color=vec(1,1,0))
        r+=dr
scene=canvas(width=800, height=600, center=vector(4,2,0))
Nj=5; Ni=20; s=0.4; v=10; R=s/4
Q1=sphere(pos=vec(0,0,0),radius=R,color=vec(1,0,0))
for j in range(Nj):
    deg=30+10*j
    t=radians(deg)
    r1=vec(cos(t),sin(t),0)*s
    line(v,Ni,s,r1)

run this code(EL-04A) in glowscript

點電荷的電力線

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V

def Eline(q1,rq1,r1):
    N=40; s=0.05
    for i in range(N):
        E,Es,V=EFV_point(q1,rq1,r1)
        dr=hat(E)*s
        arrow(pos=r1,axis=dr,shaftwidth=0.02,headwidth=0.02,color=vec(1,1,1))
        r1+=dr

q1=1.; rq1=vec(0.,0.,0.); R=0.2
scene=canvas(width=800, height=600, center=vector(0,0,0))
X=arrow(pos=vec(0,0,0),axis=vec(2,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,2),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Q1=sphere(pos=rq1,radius=R,color=vec(1,0,0))
Nr=10
s=0.1; r1=vec(1,0,0)
for jt in range(Nr):
    t=pi/Nr*jt
    for jf in range(Nr):
        f=2*pi/Nr*jf
        r1=R*vec(cos(f)*sin(t),sin(f)*sin(t),cos(t))
        Eline(q1,rq1,r1)

run this code(EL-04) in glowscript

電偶極的電力線

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V

def Eline(q1,rq1,q2,rq2,r1):
    N=40; s=0.05
    for i in range(N):
        E1,Es1,V1=EFV_point(q1,rq1,r1)
        E2,Es2,V2=EFV_point(q2,rq2,r1)
        dr=hat(E1+E2)*s
        arrow(pos=r1,axis=dr,shaftwidth=0.02,headwidth=0.02)
        r1+=dr

q1=1.; rq1=vec(0.,0.,0.); R=0.2
q2=-1.; rq2=vec(2.,0.,0.);
scene=canvas(width=800, height=600, center=vector(0,0,0))
X=arrow(pos=vec(0,0,0),axis=vec(2,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,2),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Q1=sphere(pos=rq1,radius=R,color=vec(1,0,0))
Q2=sphere(pos=rq2,radius=R,color=vec(0,0,1))
Nr=10
s=0.1; r1=vec(1,0,0)
for jt in range(Nr):
    t=pi/Nr*jt
    for jf in range(Nr):
        f=2*pi/Nr*jf
        r1=R*vec(cos(f)*sin(t),sin(f)*sin(t),cos(t))    
        Eline(q1,rq1,q2,rq2,r1)

run this code(EL-05) in glowscript

帶電直線的電力線

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V

def linesource(r):
    global NQ,q,L,dy
    E=vec(0,0,0)
    for i in range(NQ):
        rq=vec(0,-L/2+dy*i,0)
        Ei,Esi,Vi=EFV_point(q,rq,r)
        E+=Ei
    return E
    
def Eline(r1):
    N=40; s=0.05
    for i in range(N):
        E=linesource(r1)
        dr=hat(E)*s
        arrow(pos=r1,axis=dr,shaftwidth=0.02,headwidth=0.02)
        r1+=dr

q1=1.; rq1=vec(0.,0.,0.); R=0.1
q2=-1.; rq2=vec(2.,0.,0.);
scene=canvas(width=800, height=600, center=vector(0,0,0))
X=arrow(pos=vec(0,0,0),axis=vec(2,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,2),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
NQ=21;q=1/NQ; L=8; dy=L/(NQ-1); E=vec(0,0,0)
for i in range(NQ):
    rq=vec(0,-L/2+dy*i,0)
    sphere(pos=rq,radius=R,color=color.red)
Nr=11
s=0.2; r1=vec(1,0,0)
for jy in range(Nr):
    y=-5+1*jy
    for jf in range(Nr):
        f=2*pi/(Nr-1)*jf
        r1=vec(R*cos(f),y,R*sin(f))    
        Eline(r1)

run this code(EL-06) in glowscript

3. 高斯定律

單一點電荷的高斯定律

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V

def Eline(q1,rq1,r1):
    N=30; s=0.05
    for i in range(N):
        E,Es,V=EFV_point(q1,rq1,r1)
        dr=hat(E)*s
        arrow(pos=r1,axis=dr,shaftwidth=0.02,headwidth=0.02,color=vec(1,1,1))
        r1+=dr

q1=1.; rq1=vec(0.,0.,0.); R=0.2; RG=R*10
scene=canvas(width=1000, height=800, center=vector(0,0,0))
X=arrow(pos=vec(0,0,0),axis=vec(2,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,2),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Q1=sphere(pos=rq1,radius=R,color=vec(1,0,0))
G=sphere(pos=rq1,radius=RG,color=vec(1,1,0), opacity=0.2)
Nr=6; NA=Nr*Nr; dA=4*pi*RG**2/NA
s=0.1; r1=vec(1,0,0); FLUX=0
for jt in range(Nr):
    t=pi/Nr*jt
    for jf in range(Nr):
        f=2*pi/Nr*jf
        r1=R*vec(cos(f)*sin(t),cos(t),sin(f)*sin(t))
        Eline(q1,rq1,r1)
        rG=RG*vec(cos(f)*sin(t),cos(t),sin(f)*sin(t))
        dS=hat(r1)*dA
        arrow(pos=rG,axis=dS*0.5,shaftwidth=0.04,headwidth=0.08,color=vec(0,0.5,1))
        E,Es,V=EFV_point(q1,rq1,rG)
        FLUX+=dot(dS,E)
print(NA,FLUX)    
# k=1; 1/(4 pi epsilon_0)=1; 1/epsilon_0=4pi; Gauss law:FLUX=Q/epsilon_0=4 pi Q=4 pi=12.566

run this code(EL-07) in glowscript

電偶極的高斯定律

GlowScript 3.0 VPython
def EFV_point(q,rq,r):
    ke=1.
    rrq=r-rq
    rrq0=mag(rrq)
    Es=ke*q/rrq0**2
    V=ke*q/rrq0
    E=ke*q*rrq/rrq0**3
    return E,Es,V

def Eline(r1):
    N=30; s=0.1
    for i in range(N):
        E1,Es1,V1=EFV_point(q1,rq1,r1)
        E2,Es2,V2=EFV_point(q2,rq2,r1)
        E=E1+E2
        dr=hat(E)*s
        arrow(pos=r1,axis=dr,shaftwidth=0.02,headwidth=0.06,color=vec(1,1,1))
        r1+=dr
        if(mag(r1) > 1): break

global q1,rq1,q2,rq2
q1=1.; rq1=vec(-0.5,0.,0.); R=0.2; RG=R*10
q2=-1.; rq2=vec(0.5,0,0.)
scene=canvas(width=1000, height=800, center=vector(0,0,0))
X=arrow(pos=vec(0,0,0),axis=vec(2,0,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Y=arrow(pos=vec(0,0,0),axis=vec(0,2,0),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Z=arrow(pos=vec(0,0,0),axis=vec(0,0,2),shaftwidth=0.02,headwidth=0.04,color=vec(1,1,0))
Q1=sphere(pos=rq1,radius=R,color=vec(1,0,0))
Q2=sphere(pos=rq2,radius=R,color=vec(0,0,1))
G=sphere(pos=vec(0,0,0),radius=RG,color=vec(1,1,0), opacity=0.2)
Nr=6; NA=Nr*Nr; dA=4*pi*RG**2/NA
s=0.1; r1=vec(1,0,0); FLUX=0
for jt in range(Nr):
    t=pi/Nr*jt
    for jf in range(Nr):
        f=2*pi/Nr*jf
        r1=9*R*vec(cos(f)*sin(t),cos(t),sin(f)*sin(t))
        sphere(pos=r1,radius=0.3*R,color=vec(0,0,0))
        Eline(r1)
        rG=RG*vec(cos(f)*sin(t),cos(t),sin(f)*sin(t))
        dS=hat(r1)*dA
        arrow(pos=rG,axis=dS*0.5,shaftwidth=0.04,headwidth=0.08,color=vec(0,0.5,1))
        E1,Es1,V1=EFV_point(q1,rq1,rG)
        E2,Es2,V2=EFV_point(q2,rq2,rG)
        E=E1+E2
        arrow(pos=rG,axis=E*5,shaftwidth=0.04,headwidth=0.08,color=vec(1,0,0))
        FLUX+=dot(dS,E)

print(NA,FLUX)        
# k=1; 1/(4 pi epsilon_0)=1; 1/epsilon_0=4pi; Gauss law:FLUX=Q/epsilon_0=4 pi Q=4 pi=12.566

run this code(EL-08) in glowscript