2024年4月11日发(作者：)

静电场的环流为0证明

### English Answer：

The circulation of an electrostatic field is zero

because it is a conservative force field. A conservative

force field is one in which the work done by the field on a

particle moving around a closed loop is zero. This means

that the force field does not do any work on the particle,

and the particle's energy does not change as it moves

around the loop.

In an electrostatic field, the force on a particle is

given by the Coulomb force:

F = k q1 q2 / r^2。

where k is the Coulomb constant, q1 and q2 are the

charges of the two particles, and r is the distance between

the particles. The force is always directed along the line

connecting the two particles.

If we take a particle and move it around a closed loop

in an electrostatic field, the work done by the field on

the particle is given by the line integral of the force

around the loop:

W = ∮ F · dr.

where dr is the differential displacement vector.

Since the force is always directed along the line

connecting the two particles, the line integral of the

force around the loop is zero. This is because the force is

always perpendicular to the displacement vector, so the dot

product of the force and the displacement vector is zero.

Therefore, the work done by the electrostatic field on

a particle moving around a closed loop is zero. This means

that the electrostatic field is a conservative force field,

and its circulation is zero.

### 中文回答：

静电场的环流为0，因为静电场是一个保守力场。保守力场是

指场对绕闭合回路运动的粒子所做的功为0。这意味着力场不对粒

子做功，粒子在回路中运动时能量不会改变。

在静电场中，对粒子的力由库仑力给出：

F = k q1 q2 / r^2。

其中k为库仑常数，q1和q2是两个粒子的电荷，r是两个粒子

之间的距离。该力始终沿连接两个粒子的直线方向。

如果我们取一个粒子并将其在静电场中沿着闭合回路移动，则

场对粒子所做的功由力在回路周围的线积分给出：

W = ∮ F · dr.

其中dr是微分位移矢量。

由于力始终沿连接两个粒子的直线方向，因此力在回路周围的

线积分等于零。这是因为力始终垂直于位移矢量，因此力和位移矢

量的点积为零。

因此，静电场对绕闭合回路运动的粒子所做的功为零。这意味

着静电场是一个保守力场，其环流为0。