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时间:2021-08-05
《电磁现象的普遍规律.docx》由会员上传分享,免费在线阅读,更多相关内容在教育资源-天天文库。
1、第一章电磁现象的普遍规律1.根据算符△的微分性与矢量性,推导下列公式:B)K))Bq(AB)=B(vA(B)AAd解:矢量性为a(bc)=b(ca)=c(ab)cx(am,)=(bc)a—(Ca)b②d414Jd(am,)Mc=(ca)b-(cb)a③微商性■I・dd/daE^db—(ab)=——b+a—④dtdtdt।・।公d"「dad-db—(aMb)=—Mb+aM——⑤dtdtdt由②得1・・二」iBcx(VxA)=$(BcA)-(BcV)A⑥人父①mB)=V(AcB)-(Ac$)B⑦⑥+⑦得景(5A)R0B)—(BcA)r(AcB)-
2、
3、(Bc)A(Ac)B因为,(AB)='1(AcB
4、)(KbJ)二上式得・d・一」(AB)=Bc仆A)Ac仆B)(BcK)A(Ac』)B令B=A得'、A=2A(1■'A)A)2(A^)AA仆A)"”A2-(A^)A2.设W是空间坐标x,v,z的函数,证明:df1kf(u)=——-udu大A^(U)=Ud-duA(u)=3ud-du解:①vf(u).-f(u)ex-yf(u)ey-f(u)ezdf(u)fu.df(u)du;xQ:u・df(u)一eyezduydu;zdf(u)du.xdf(u)/Fu・:u-(—exeydufxjynnnA(u)二—Ax—Ay一Az;x;y;zdAx::udAyfudAz::u-I--du;xdu、.dA=
5、%u—du:ydu;z'、、A(u)=exex6、。,(r=0)rrrrrrr(最后一式在r=0点不成立,见第二章第五节)⑵求▽/KF"*V)r^,V(a,,),▽,[Esin(k$]及炉父[Esin(kr)],其中a,k及E均为常矢量。解:⑴一:r■:r□.:r、r二0一eyez—:xcyczMx-x,片y-yex■’2'2'2ey1‘2'2‘2(x-x)(y-y)(z-z)(x-x)(y-y)(z-z)z-zx(x-x,)2(y-y,)2(z-z,)2「r=q3r,'''、,Yx-x)・-(y-y)■-(z-z)二e-e,ez^x222y222z222(x-x)(y-y)(z-z)(x-x)(y-y)(z-z)(x-x)(y-y)(z-z7、)dr=I-ri11-:1-''一二一(一)ex,—(-)ey-—(-)ezr-:r.,yez)二zr二xrcyrczr-1/::r.:r・「(一ex一er二xcy3r―r(一)ex-r(—)ey-r(—)ez.xryr;zr-1:r-.:r..:r•=-2('ex'~ey'~ez)rFx7;z1.-3,■=f0—rrr34-3rd”一rrr=0r或这样证明'、、33r一」r=0(梯度的旋度恒为零)r於rrr。1、r.—r—3r■1=F(一r)飞rrr=0(r=0)=▽rr’1d(3r)一3一r(7d1r)-(-3)rr__-、'=0(r=0)r二-(X-X‛)一(y_yj—(z-zj.x=8、3ex.:xx-xeyyy-yz~z)r-(ax-ayjx—IC,c■一az—)r.y;zrjrfr二ax一■ay—az-:xcy:z一一・=a*exayeyazez二a(注!这里如果没有引入张量。可以采用分量的方法进行证明第一项,如下)frax一:x=ax-[ex(x-x)ey(y-y)ez(z—z)]=a*ex同理,Fr-az—=azG,二z故(a,▽)r=axex+ayey+azez=a(ar)d4d4d4dd=aGr)(a))r・r仆a)(r、)a二,0(aD:+00■L・二(a、)r.二a、[ETsin(H)]:sin(^r)]尽sin(mr)%,e'..,'..,'一小sin(k9、x(x—x)ky(y-y)kz(z-z))]=Eoxkxcos[kx(x—x)ky(y-y)kz(z-z)]EoykyCos[kx(x-x)ky(y-y)kz(z-z)]'''Eozkzcos[kx(x-x)ky(y-y)kz(z-z)]二(EoxkxEoykyE°zkz)cos[kx(x-x)ky(y-y)kz(z-z)]=cos('、[Eosin(kr)]=「.sin(kr)]Eosin■=[
6、。,(r=0)rrrrrrr(最后一式在r=0点不成立,见第二章第五节)⑵求▽/KF"*V)r^,V(a,,),▽,[Esin(k$]及炉父[Esin(kr)],其中a,k及E均为常矢量。解:⑴一:r■:r□.:r、r二0一eyez—:xcyczMx-x,片y-yex■’2'2'2ey1‘2'2‘2(x-x)(y-y)(z-z)(x-x)(y-y)(z-z)z-zx(x-x,)2(y-y,)2(z-z,)2「r=q3r,'''、,Yx-x)・-(y-y)■-(z-z)二e-e,ez^x222y222z222(x-x)(y-y)(z-z)(x-x)(y-y)(z-z)(x-x)(y-y)(z-z
7、)dr=I-ri11-:1-''一二一(一)ex,—(-)ey-—(-)ezr-:r.,yez)二zr二xrcyrczr-1/::r.:r・「(一ex一er二xcy3r―r(一)ex-r(—)ey-r(—)ez.xryr;zr-1:r-.:r..:r•=-2('ex'~ey'~ez)rFx7;z1.-3,■=f0—rrr34-3rd”一rrr=0r或这样证明'、、33r一」r=0(梯度的旋度恒为零)r於rrr。1、r.—r—3r■1=F(一r)飞rrr=0(r=0)=▽rr’1d(3r)一3一r(7d1r)-(-3)rr__-、'=0(r=0)r二-(X-X‛)一(y_yj—(z-zj.x=
8、3ex.:xx-xeyyy-yz~z)r-(ax-ayjx—IC,c■一az—)r.y;zrjrfr二ax一■ay—az-:xcy:z一一・=a*exayeyazez二a(注!这里如果没有引入张量。可以采用分量的方法进行证明第一项,如下)frax一:x=ax-[ex(x-x)ey(y-y)ez(z—z)]=a*ex同理,Fr-az—=azG,二z故(a,▽)r=axex+ayey+azez=a(ar)d4d4d4dd=aGr)(a))r・r仆a)(r、)a二,0(aD:+00■L・二(a、)r.二a、[ETsin(H)]:sin(^r)]尽sin(mr)%,e'..,'..,'一小sin(k
9、x(x—x)ky(y-y)kz(z-z))]=Eoxkxcos[kx(x—x)ky(y-y)kz(z-z)]EoykyCos[kx(x-x)ky(y-y)kz(z-z)]'''Eozkzcos[kx(x-x)ky(y-y)kz(z-z)]二(EoxkxEoykyE°zkz)cos[kx(x-x)ky(y-y)kz(z-z)]=cos('、[Eosin(kr)]=「.sin(kr)]Eosin■=[
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