📚 CIE A-Level Physics: Radial Electric Field Strength Formula and Graph Analysis | CIE A-Level 物理：径向电场强度公式与图像分析
In CIE A-Level Physics, the radial electric field is one of the most important electrostatic concepts. It is produced by an isolated point charge or by a uniformly charged sphere at points outside the sphere. Understanding its field strength formula E = Q/(4πε₀r²) and the corresponding graphs is essential for Paper 1, Paper 2, and Paper 4 questions.
在 CIE A-Level 物理中，径向电场是最重要的静电学概念之一。它由孤立点电荷或均匀带电球体在球外空间产生。理解其场强公式 E = Q/(4πε₀r²) 以及相应图像，对 Paper 1、Paper 2 和 Paper 4 的题目都至关重要。
1. What Is a Radial Electric Field? | 什么是径向电场？
A radial electric field is a field in which the field lines point directly towards or away from a central point charge. For a positive charge, the lines point radially outward; for a negative charge, they point radially inward. The field strength depends only on the distance r from the centre of the charge.
径向电场是一种电场线直接指向或背离中心点电荷的电场。对于正电荷，电场线径向向外；对于负电荷，电场线径向向内。场强只取决于到电荷中心的距离 r。
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For an isolated point charge, the field exists at all points around the charge.
对于孤立点电荷，电场存在于电荷周围的所有点。
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For a uniformly charged conducting sphere, the formula E = Q/(4πε₀r²) applies only outside the sphere, where r is measured from the centre.
对于均匀带电的导体球，公式 E = Q/(4πε₀r²) 只适用于球外，其中 r 从球心算起。
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Inside a charged conducting sphere, the electric field strength is zero in electrostatic equilibrium.
在静电平衡下，带电导体球内部的电场强度为零。
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The spacing of field lines indicates the strength of the field: closer lines mean a stronger field.
电场线的疏密表示场强大小：电场线越密，场越强。
2. Electric Field Strength and Coulomb’s Law | 电场强度与库仑定律
The electric field strength E at a point is defined as the force per unit positive test charge placed at that point.
电场强度 E 定义为作用于该点的单位正检验电荷所受的力。
E = F / q
According to Coulomb’s law, the electric force between two point charges Q and q separated by distance r is:
根据库仑定律，两个相距 r 的点电荷 Q 与 q 之间的静电力为：
F = Qq / (4πε₀r²)
Substituting this into the definition of field strength gives the radial field strength formula:
将上式代入场强定义式，得到径向场强公式：
E = Q / (4πε₀r²) = kQ / r²
Here, Q is the source charge in coulombs, r is the distance from the centre of the charge in metres, ε₀ is the permittivity of free space (8.85 × 10⁻¹² C² N⁻¹ m⁻²), and k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻².
式中，Q 为源电荷（单位库仑），r 为到电荷中心的距离（单位米），ε₀ 为真空介电常数（8.85 × 10⁻¹² C² N⁻¹ m⁻²），k = 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻²。
3. Key Features of the Radial Field Formula | 径向场强公式的关键特征
The formula E = kQ/r² has several features that CIE often tests.
公式 E = kQ/r² 有若干 CIE 常考的关键特征。
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E is inversely proportional to the square of the distance: E ∝ 1/r². Doubling r makes E four times smaller.
E 与距离的平方成反比：E ∝ 1/r²。r 加倍时，E 减小为原来的四分之一。
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E is proportional to the source charge Q. If Q is doubled, E is doubled at the same distance.
E 与源电荷 Q 成正比。同一距离处，Q 加倍，E 也加倍。
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E does not depend on the test charge. The field is a property of the source charge and the surrounding space.
E 与检验电荷无关。电场是源电荷及其周围空间的性质。
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The direction of E is radial. For Q > 0, E points away from the charge; for Q < 0, E points towards the charge.
E 的方向沿径向。当 Q > 0 时，E 背离电荷；当 Q < 0 时，E 指向电荷。
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Because E is a vector, two fields at the same point must be added by vector addition, not by simple scalar addition.
由于 E 是矢量，同一点处的两个电场必须用矢量合成，而不是简单的标量相加。
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