高斯定律

科学是一门引人入胜的学科，充满了引人入胜的信息。对科学原理及其相关学科的深入研究越多，获得的知识和信息就越多。分析电荷、表面和电通量问题的高斯定律就是这样一个研究课题。让我们更多地了解法律及其运作方式，以便我们理解法律的方程式。

什么是高斯定律？

根据高斯定律，封闭表面的总电通量等于所包含的电荷除以介电常数。通过将电场乘以投影在垂直于电场的平面中的表面面积来计算给定区域中的电通量。

The total flux associated with a closed surface equals 1 ⁄ ε₀ times the charge encompassed by the closed surface, according to the Gauss law.

∮ E ds = q ⁄ ε₀

例如，将点电荷“q”放在边为“a”的立方体内。根据高斯定律，立方体每个面的通量现在为 q ⁄ 6ε ₀ 。

电场是理解电的最基本的概念。通常，表面的电场是使用库仑方程计算的，但是，需要理解高斯定律的概念来计算闭合表面中的电场分布。它描述了电荷如何被封闭在封闭表面中或电荷如何存在于封闭的封闭表面中。

高斯定律公式

根据高斯定理，封闭表面所包围的总电荷与表面所包围的总通量成正比。因此，表面所包含的总电荷 Q 为： 如果 ε ₀是电常数，而 φ 是总通量。

Q = ϕ ε₀

The formula of Gauss law is given by:

ϕ = Q ⁄ ε₀

where,

• ε₀ is electric constant,
• Q is total charge within a given surface, and
• ϕ is flux enclosed by surface.

高斯定理

封闭表面所包含的体积中的净电荷与通过封闭表面的净通量完全成正比。

φ = E dA = q_净⁄ ε ₀

高斯定理用简单的术语将电场线（通量）的“流动”与封闭表面内的电荷联系起来。如果表面不包含电荷，则净电流保持为 0。进入表面的电场线的数量等于离开表面的场线的数量。

高斯定理陈述还给出了一个重要的推论：

来自任何封闭表面的电通量仅由表面所包围的电场的源和汇引起。电通量不受表面外任何电荷的影响。此外，只有电荷可以作为电场源或汇。例如，变化的磁场不能充当电场源或电场。

因为它包含净电荷，所以左侧表面的净流量不为零。因为右手表面不包含任何电荷，净流量为零。高斯定律只不过是库仑定律的重复。库仑定律很容易通过将高斯定理应用于被球体包围的点电荷来获得。

高斯定律的应用

• 在介电常数为 K 的介质中，平面带电导体附近的电场强度 E = σ ⁄ K ε ₀ 。当电介质为空气时，E _air = σ ⁄ ε ₀ 。
• 在无限电荷线的情况下，距离为 'r'，E = (1 ⁄ 4 × π r ε ₀ ) (2π ⁄ r) = λ ⁄ 2π r ε ₀ ，其中 λ 是线性电荷密度。
• 在冷凝器中，两个平行板之间的场为 E = σ ⁄ ε ₀ ，其中 σ 是表面电荷密度。
• 电荷平面附近的电场强度为 E = σ ⁄ 2K ε0，其中 σ 是表面电荷密度。

示例问题

问题 1：在 x 方向，有一个大小为 E = 50 N⁄C 的均匀电场。使用高斯定理计算该场在 yz 平面中边缘为 5 cm 的平面正方形区域的通量。假设法线沿正 x 轴为正。

解决方案：

Given:

Electric field, E = 50 N⁄C

Edge length of square, a = 5 cm = 0.05 m

The flux of the field across a plane square, ϕ = ∫ E cosθ ds

As the normal to the area points along the electric field, θ = 0.

Also, E is uniform so, Φ = E ΔS = (50 N⁄C) (0.05 m)² = 0.125 N m² C^-1.

Hence, the flux of the given field is 0.125 N m² C^-1.

问题2：我们如何为不同的情况选择合适的高斯曲面？

解决方案：

In order to select an acceptable Gaussian Surface, we must consider the fact that the charge-to-dielectric constant ratio is supplied by a (two-dimensional) surface integral over the charge distribution’s electric field symmetry.

We’ll need to know about three potential scenarios.

• When the charge distribution is spherically symmetric, it is called spherical.
• When the charge distribution is cylindrically symmetric, it is called cylindrical.
• When the charge distribution exhibits translational symmetry along a plane, it is called a pillbox.

Depending on where we want to compute the field, we may determine the size of the surface. The Gauss theorem is useful for determining the direction of a field when there is symmetry, as it informs us how the field is directed.

问题3：如何用高斯定律求电场？

解决方案：

Normally, the Gauss law is employed to calculate the electric field of symmetric charge distributions. When using this law to solve the problem of the electric field, there are numerous processes required. The following are the details:

1. First, we must determine the charge distribution’s spatial symmetry.
2. The next step is to select a proper Gaussian surface that has the same symmetry as the charge distribution. Its ramifications must also be determined.
3. Calculate the flux across the surface by evaluating the integral ϕ_s E over the Gaussian surface.
4. Calculate the amount of charge contained within the Gaussian surface.
5. Calculate the charge distribution’s electric field.

However, in order to determine the electric field, pupils must remember the three forms of symmetry. The following are the several forms of symmetry:

• Symmetry on a sphere
• Symmetry in a cylindrical shape
• Symmetry on a plane

问题 4：有 q1、q2 和 q3 三个电荷，其电荷 4 C、7 C 和 2 C 包围在一个表面中。求曲面包围的总通量。

解决方案：

Total charge Q,

Q = q₁ + q₂ + q₃

= 4 C + 7 C + 2 C

= 13 C

The total flux, ϕ = Q ⁄ ε₀

ϕ = 13 C ⁄ (8.854×10⁻¹² F ⁄ m)

ϕ = 1.468 N m² C^-1

Therefore, the total flux enclosed by the surface is 1.584 N m² C^-1.

问题5：高斯定理的微分形式是什么？

解决方案：

The electric field is related to the charge distribution at a certain location in space by the differential version of Gauss law. To clarify, according to the law, the electric field’s divergence (E) is equal to the volume charge density (ρ) at a given position. It’s written like this:

ΔE = ρ ⁄ ε₀

Here, ε₀ is the permittivity of free space.