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level,第三节,高斯定理,一、电力线,电力线（,E,）线：在电场中画一组曲线，,曲线上每一点的切线方向与该点的电场方向,一致，这一组曲线称为电力线。,一、电力线,电力线（,E,）线：在电场中画一组曲线，,曲线上每一点的切线方向与该点的电场方向,一致，这一组曲线称为电力线。,E,一、电力线,电力线（,E,）线：在电场中画一组曲线，,曲线上每一点的切线方向与该点的电场方向,一致，这一组曲线称为电力线。,为了定量地描写电场，对电力线的画法,作如下的规定：在电场中任一点处，通过垂,直于电场强度,E,单位面积的电力线数等于该,点的电场强度的数值。,E,d,S,E,一、电力线,点电荷的电力线,正电荷,负电荷,+,+,一对等量异号电荷的电力线,一对等量正点电荷的电力线,+,+,一对异号不等量点电荷的电力线,2,q,q,+,带电平行板电容器的电场,+,+,+,+,+,+,+,+,+,静电电力线性质：,1、起源于正电荷或无穷远处,静电电力线性质：,1、起源于正电荷或无穷远处,终止于负电荷或无穷远处,静电电力线性质：,1、起源于正电荷或无穷远处,终止于负电荷或无穷远处,静电电力线性质：,2、不形成闭合曲线、不中断,1、起源于正电荷或无穷远处,终止于负电荷或无穷远处,静电电力线性质：,2、不形成闭合曲线、不中断,任何两条电力线不会相交,二、电通量、高斯定理,e,=,E,S,S,E,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,e,=,E,S,S,E,S,S,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,e,=,E,S,=,E,S,cos,S,E,S,S,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,e,=,E,S,=,E,S,cos,=,E,S,.,S,E,S,S,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,e,=,E,S,=,E,S,cos,=,E,S,.,.,e,=,d,E,d,S,S,E,S,S,d,S,E,二、电通量、高斯定理,e,=,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,e,=,E,S,=,E,S,cos,=,E,S,.,.,e,=,d,E,d,S,E,=,cos,d,S,S,E,S,S,d,S,E,二、电通量、高斯定理,E,.,e,=,s,E,S,e,=,E,S,S,E,1、电通量：通过某一面积,的电力线线数,e,=,E,S,=,E,S,cos,=,E,S,.,.,e,=,d,E,d,S,E,=,cos,d,S,e,=,d,S,S,E,S,S,d,S,E,二、电通量、高斯定理,2、高斯定理,从点电荷特例引出此定理,2、高斯定理,+,r,从点电荷特例引出此定理,q,2、高斯定理,+,r,从点电荷特例引出此定理,q,2、高斯定理,+,r,从点电荷特例引出此定理,E,q,2、高斯定理,+,r,从点电荷特例引出此定理,d,S,E,q,2、高斯定理,+,r,从点电荷特例引出此定理,s,E,.,d,S,d,S,E,q,2、高斯定理,+,r,从点电荷特例引出此定理,s,E,.,d,S,=,s,d,S,cos,0,0,d,S,E,q,2、高斯定理,2,r,4,q,O,+,+,r,从点电荷特例引出此定理,s,E,.,d,S,=,s,d,S,cos,0,0,=,d,S,E,q,+,2、高斯定理,s,d,S,2,r,4,q,O,2,r,4,q,O,+,+,r,从点电荷特例引出此定理,s,E,.,d,S,=,s,d,S,cos,0,0,=,=,q,d,S,E,q,+,2、高斯定理,s,d,S,2,r,4,q,O,2,r,4,q,O,+,+,O,+,r,从点电荷特例引出此定理,s,E,.,d,S,=,s,d,S,cos,0,0,=,=,q,d,S,E,q,讨论：,1,. 若,+,反, 上式积分值为负值。,方向相,d,S,的方向与,E,为负值，则,q,2、高斯定理,s,d,S,2,r,4,q,O,2,r,4,q,O,+,+,O,+,r,从点电荷特例引出此定理,s,E,.,d,S,=,s,d,S,cos,0,0,=,=,q,d,S,E,q,讨论：,1,. 若,+,反, 上式积分值为负值。,上式中的,q,应理解为代数值。,方向相,d,S,的方向与,E,为负值，则,q,2、高斯定理,s,d,S,2,r,4,q,O,2,r,4,q,O,+,+,O,s,E,.,d,S,=,q,O,/,2,. 此式的意义是通过闭合曲面的电力线,条数等于面内的电荷数除以真空介电常数。,s,E,.,d,S,=,q,O,/,2,. 此式的意义是通过闭合曲面的电力线,条数等于面内的电荷数除以真空介电常数。,3,. 若电荷在面外，则此积分值为,0,。因,为有几条电力线进面内必然有同样数目的电,力线从面内出来。,s,E,.,d,S,=,q,+,q,O,/,2,. 此式的意义是通过闭合曲面的电力线,条数等于面内的电荷数除以真空介电常数。,3,. 若电荷在面外，则此积分值为,0,。因,为有几条电力线进面内必然有同样数目的电,力线从面内出来。,4,. 若封闭面不是球面，则积分值不变。,s,E,.,d,S,=,q,+,q,q,O,/,5,. 若面内有若干个电荷，则积分值为：,5,. 若面内有若干个电荷，则积分值为：,s,E,.,d,S,=,i,q,O,/,5,. 若面内有若干个电荷，则积分值为：,s,E,.,d,S,=,高斯定理,: 在静电场中，通过任意封闭,曲面电力线矢量的通量，等于面内所包围的,电荷代数和除以真空中的介电常数。,i,q,O,/,D,=,E,=,r,o,E,电位移的定义：,D,=,E,=,r,o,E,电位移的定义：,高斯定理,: 在静电场中，通过任意封闭,曲面电位移线矢量的通量，等于面内所包围的自由电荷代数。,D,=,E,=,r,o,E,电位移的定义：,s,.,d,S,=,高斯定理,: 在静电场中，通过任意封闭,曲面电位移线矢量的通量，等于面内所包围的自由电荷代数。,oi,q,D,第四节,高斯定理的应用,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,q,一、,均匀带电球面的电场,R,（1）,r,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,q,一、,均匀带电球面的电场,R,（1）,r,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,一、,均匀带电球面的电场,R,（1）,r,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,r,一、,均匀带电球面的电场,R,0,0,一、,均匀带电球面的电场,（1）,r,R,s,E,.,d,S,=,s,E,d,S,cos,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,r,R,0,0,一、,均匀带电球面的电场,（1）,r,R,s,E,.,d,S,=,=,s,E,d,S,cos,E,s,d,S,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,r,R,=,0,0,一、,均匀带电球面的电场,（1）,r,R,s,E,.,d,S,=,=,s,E,d,S,cos,E,s,d,S,E,2,r,4,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,r,R,=,0,0,一、,均匀带电球面的电场,（1）,r,R,s,E,.,d,S,=,=,s,E,d,S,cos,E,s,d,S,E,2,r,4,=,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,r,q,i,/,O,R,=,0,0,一、,均匀带电球面的电场,（1）,r,R,s,E,.,d,S,=,=,s,E,d,S,cos,E,s,d,S,E,2,r,4,=,0,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,=,E,q,高斯面,r,q,i,/,O,R,=,R,（2）,r,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,q,R,（2）,r,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,E,q,R,（2）,r,R,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,R,（2）,r,R,s,E,.,d,S,=,E,2,r,4,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,r,E,q,高斯面,R,（2）,r,R,s,E,.,d,S,=,E,2,r,4,=,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,q/,0,r,E,q,高斯面,R,（2）,r,R,s,E,.,d,S,=,E,2,r,4,=,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,q/,0,r,E,q,高斯面,=,E,2,r,4,q,O,R,（2）,r,R,s,E,.,d,S,=,E,2,r,4,=,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,q/,0,r,E,q,r,E,2,r,1,0,R,高斯面,=,E,2,R,4,q,O,2,r,4,q,O,2,. 均匀带电球体的电场。体电荷密度为,2,. 均匀带电球体的电场。体电荷密度为,R,O,2,. 均匀带电球体的电场。体电荷密度为,（1）,r,R,R,O,2,. 均匀带电球体的电场。体电荷密度为,（1）,r,R,R,E,O,2,. 均匀带电球体的电场。体电荷密度为,（1）,r,R,R,r,E,O,高斯面,（1）,r,R,s,E,d,S,=,E,2,r,4,R,r,E,O,.,二、,均匀带电球体的电场。体电荷密度为,高斯面,=,（1）,r,R,s,E,d,S,=,E,2,r,4,3,r,4,3,R,r,E,O,.,二、,均匀带电球体的电场。体电荷密度为,/,O,高斯面,=,（1）,r,R,E,2,r,4,s,E,d,S,=,E,2,r,4,3,r,4,3,=,3,R,4,3,E,=,R,3,3,r,2,O,R,r,R,r,E,E,O,.,二、,均匀带电球体的电场。体电荷密度为,高斯面,/,E,O,=,r,3,O,/,O,=,Q,0,O,=,Q,4,0,r,2,均匀带电球体电力线及电场强度分布曲线,R,E,O,均匀带电球体电力线及电场强度分布曲线,E,r,O,R,O,r,3,R,3,3,r,2,O,三、,均匀带电无限大平面的电场,三、,均匀带电无限大平面的电场,E,E,三、,均匀带电无限大平面的电场,E,E,三、,均匀带电无限大平面的电场,高,斯,面,s,E,.,d,S,E,E,S,三、,均匀带电无限大平面的电场,s,E,.,d,S,=,侧,E,.,d,S,E,E,S,三、,均匀带电无限大平面的电场,s,E,.,d,S,=,侧,E,.,d,S,左底,E,.,d,S,+,E,E,S,三、,均匀带电无限大平面的电场,s,E,.,d,S,=,侧,E,.,d,S,左底,E,.,d,S,右底,E,.,d,S,+,+,E,E,S,三、,均匀带电无限大平面的电场,=,s,E,.,d,S,=,侧,E,.,d,S,左底,E,.,d,S,右底,E,.,d,S,+,+,0,E,E,S,三、,均匀带电无限大平面的电场,=,=,s,E,.,d,S,=,侧,E,.,d,S,左底,E,.,d,S,右底,E,.,d,S,+,+,E,S,+,E,S,0,E,E,S,三、,均匀带电无限大平面的电场,=,=,s,E,.,d,S,=,=,侧,E,.,d,S,左底,E,.,d,S,右底,E,.,d,S,+,+,S,E,S,+,E,S,0,E,E,S,三、,均匀带电无限大平面的电场,O,=,=,s,E,.,d,S,=,=,侧,E,.,d,S,左底,E,.,d,S,右底,E,.,d,S,+,+,S,E,S,+,E,S,0,E,=,2,O,E,E,S,三、,均匀带电无限大平面的电场,O,四、均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,四、均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,（1）,r,R,四、均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,（1）,r,R,四、均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,E,（1）,r,R,四、均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,高,斯,面,E,（1）,r,R,四、均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,E,=,r,2,l,0,E,（1）,r,R,4,. 均匀带电圆柱面的电场。,沿轴线方向单位长度带电量为,E,=,r,2,l,0,E,=,0,.,.,.,E,高,斯,面,s,E,.,d,S,E,高,斯,面,s,E,.,d,S,=,侧,E,.,d,S,E,高,斯,面,s,E,.,d,S,=,侧,E,.,d,S,上底,E,.,d,S,下底,E,.,d,S,+,+,E,高,斯,面,=,s,E,.,d,S,=,侧,E,.,d,S,上底,E,.,d,S,下底,E,.,d,S,+,+,=,0,0,E,高,斯,面,=,s,E,.,d,S,=,侧,E,.,d,S,上底,E,.,d,S,下底,E,.,d,S,+,+,=,0,0,E,高,斯,面,O,=,E,r,2,l,=,l,=,s,E,.,d,S,=,侧,E,.,d,S,上底,E,.,d,S,下底,E,.,d,S,+,+,=,0,0,E,高,斯,面,E,=,2,O,O,=,E,r,2,l,=,l,r,
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