📚 Electric Field Strength for a Radial Field | 径向电场的电场强度
In A-Level Physics, a radial electric field is the field around a point charge or a charged sphere. Understanding how electric field strength E varies with distance is essential for solving problems on Coulomb’s law, electric potential and the motion of charged particles.
在 A-Level 物理中，径向电场是点电荷或带电球体周围的电场。理解电场强度 E 如何随距离变化，对于解决库仑定律、电势以及带电粒子运动的问题至关重要。
1. Defining Electric Field Strength | 电场强度的定义
Electric field strength E at a point is defined as the force per unit positive charge placed at that point. It is given by:
电场强度 E 定义为放置在该点的单位正电荷所受的力。其公式为：
E = F / q
where F is the electric force on a small positive test charge q. The SI unit of E is newton per coulomb, N C⁻¹, which is equivalent to volt per metre, V m⁻¹.
其中 F 是作用于小正检验电荷 q 的电场力。E 的 SI 单位是牛每库仑，即 N C⁻¹，也等价于伏特每米，V m⁻¹。
Because E is defined with a positive test charge, the direction of E is the direction of the force on a positive charge. If the source charge is negative, the field points toward it.
由于 E 是用正检验电荷定义的，因此 E 的方向就是正电荷所受力的方向。如果源电荷为负，电场方向指向该电荷。
2. Radial Field Around a Point Charge | 点电荷周围的径向电场
For a point charge Q, the electric field at a distance r from the centre of the charge is radial. Its magnitude is given by Coulomb’s law for the field:
对于点电荷 Q，距离电荷中心 r 处的电场是径向的。其大小由电场形式的库仑定律给出：
E = Q / (4πε₀ r²)
Here ε₀ is the permittivity of free space, ε₀ = 8.85 × 10⁻¹² F m⁻¹. The field points directly away from a positive Q and directly toward a negative Q.
这里 ε₀ 是真空介电常数，ε₀ = 8.85 × 10⁻¹² F m⁻¹。对于正电荷 Q，电场方向沿径向向外；对于负电荷 Q，电场方向沿径向向内。
This formula is valid outside a uniformly charged sphere if r is measured from the centre of the sphere. Inside a conductor in electrostatic equilibrium, the field is zero.
如果 r 是从均匀带电球体的中心测量，该公式在球外成立。处于静电平衡的导体内部电场为零。
3. Vector Nature and Direction | 矢量性质与方向
Electric field strength is a vector. When several point charges are present, the resultant field at a point is the vector sum of the fields due to each individual charge.
电场强度是矢量。当存在多个点电荷时，某点的合电场是各个电荷单独产生的电场的矢量和。
For a single positive point charge, the field direction is radially outward from the charge. For a single negative point charge, it is radially inward. The magnitude depends only on the distance r, so the field has spherical symmetry.
对于单个正点电荷，电场方向从电荷沿径向向外。对于单个负点电荷，电场方向沿径向向内。场的大小仅取决于距离 r，因此电场具有球对称性。
To calculate a resultant radial field, you must add the electric field vectors using components. Do not simply add magnitudes unless the fields act along the same straight line.
计算合径向电场时，必须按分量进行矢量加法。除非各电场沿同一直线作用，否则不能简单地把大小相加。
4. Field Lines for a Radial Field | 径向电场的电场线
Electric field lines show the direction of the electric field. For a positive point charge, the field lines are straight lines directed radially outward, as if they originate from the charge.
电场线表示电场的方向。对于正点电荷，电场线是从电荷出发、沿径向向外的直线。
For a negative point charge, the field lines are straight lines directed radially inward, as if they terminate on the charge. The spacing of the lines increases with distance, which indicates that the field strength decreases.
对于负点电荷，电场线是沿径向向内的直线，好像终止于该电荷。电场线的间距随距离增大而增大，这表明电场强度在减小。
Field lines never cross. The number of lines per unit area perpendicular to the field is proportional to the field strength. In a radial field, the same number of lines spreads over a larger spherical surface area 4πr², so the field strength falls as 1/r².
电场线永不相交。垂直于电场的单位面积上的电场线数量与电场强度成正比。在径向电场中，同样数量的电场线分布在更大的球面面积 4πr² 上，因此电场强度按 1/r² 减弱。
5. Inverse Square Law | 平方反比定律
The relationship E ∝ 1/r² is called an inverse square law. If the distance from a point charge doubles, the electric field strength becomes one quarter of its original value.
关系式 E ∝ 1/r² 称为平方反比定律。如果到点电荷的距离加倍，电场强度将变为原来的四分之一。
If the distance triples, the field strength becomes one ninth. This rapid fall-off is a key feature of radial fields and is often tested in CIE exam questions.
如果距离变为三倍，电场强度变为九分之一。这种快速衰减是径向电场的关键特征，也是 CIE 考试中常考的内容。
The inverse square law arises because the electric field is spread over the surface area of a sphere. Since the surface area of a sphere is 4πr², doubling r multiplies the area by 4, so the field per unit area decreases by a factor of 4.
平方反比定律来源于电场分布在球面上。因为球面积是 4πr²，r 加倍会使面积乘以 4，因此单位面积上的电场减小为原来的四分之一。
6. Comparing Radial and Uniform Fields | 径向电场与匀强电场的比较
A uniform electric field, such as the field between two parallel charged plates, has a constant electric field strength. In a radial field, E changes with position.
匀强电场（例如两块平行带电板之间的电场）具有恒定的电场强度。在径向电场中，E 随位置变化。
The table below summarises the main differences:
下表总结了主要区别：
| Property | Radial field | Uniform field |
|---|---|---|
| Field strength | E = Q/(4πε₀r²), varies with r | E = V/d, constant |
| Field lines | Radial, spacing changes | Parallel, equally spaced |
| Direction | Depends on position | Same everywhere |
In a uniform field, E can be found from the potential difference V between plates separated by distance d using E = V/d. In a radial field, there is no single value of E for the whole space.
在匀强电场中，E 可以由相距 d 的两板间电势差 V 通过 E = V/d 求得。在径向电场中，整个空间不存在单一的 E 值。
7. Superposition of Radial Fields | 径向电场的叠加
When two or more point charges produce fields, the electric field at a point is the vector sum of the individual fields. This is the principle of superposition.
当两个或多个点电荷产生电场时，某点的电场是各个单独电场的矢量和。这就是叠加原理。
For charges along a straight line, the resultant field at a point on the line can be found by adding fields with appropriate signs. For example, the field due to a positive charge might be taken as one direction, while the field due to a negative charge might be taken as the opposite direction.
对于沿同一直线的电荷，可以用适当的正负号相加来求该直线上某点的合电场。例如，正电荷产生的电场可取某一方向，负电荷产生的电场取相反方向。
For off-axis points, you should resolve the field vectors into horizontal and vertical components and then combine them. A common exam task is to find the neutral point where the resultant electric field is zero.
对于不在轴线上的点，应将电场矢量分解为水平和竖直分量后再合成。常见的考题是求合电场为零的中性点。
8. Relation Between E and Electric Potential | 电场强度与电势的关系
Electric field strength and electric potential are closely related. In a radial field, the potential V at a distance r from a point charge Q is:
电场强度与电势密切相关。在径向电场中，距离点电荷 Q 为 r 处的电势 V 为：
V = Q / (4πε₀ r)
The electric field strength is the negative gradient of the potential. Along the radial direction, this gives:
电场强度是电势的负梯度。沿径向方向，这给出：
E = -dV/dr
Differentiating V = Q/(4πε₀r) with respect to r gives E = Q/(4πε₀r²), which agrees with the radial field formula. The negative sign shows that E points in the direction of decreasing potential.
对 V = Q/(4πε₀r) 求关于 r 的导数，得到 E = Q/(4πε₀r²)，与径向电场公式一致。负号表示 E 指向电势降低的方向。
9. Force on a Charge in a Radial Field | 径向电场中电荷所受的力
A charge q placed in an electric field experiences a force given by F = qE. In a radial field around a point charge Q, the force on a test charge q at distance r has magnitude:
置于电场中的电荷 q 会受到力 F = qE。在点电荷 Q 周围的径向电场中，距离 r 处检验电荷 q 所受的力大小为：
F = qQ / (4πε₀ r²)
This is Coulomb’s law. If q and Q
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