📚 Electric Potential | 电势
Electric potential is one of the most important but often misunderstood ideas in A-Level electromagnetism. It allows us to describe electric fields in terms of energy per unit charge, and it provides a powerful scalar tool for solving problems involving point charges, parallel plates and conductors. A clear understanding of potential difference is also essential for electric circuits, electron acceleration and atomic physics.
电势是 A-Level 电磁学中重要但常被误解的概念之一。它让我们能够用单位电荷的能量来描述电场，并为求解点电荷、平行板及导体问题提供了一个强大的标量工具。理解电势差对电路、电子加速以及原子物理同样至关重要。
1. Defining Electric Potential | 电势的定义
The electric potential V at a point in an electric field is defined as the work done per unit positive charge in bringing a small positive test charge from infinity to that point, without any change in kinetic energy. In other words, potential is a measure of the electric potential energy that a unit positive charge would have at that location.
电场中某点的电势 V 定义为单位正电荷从无穷远处移到该点过程中外力所做的功，且过程中电荷的动能不发生变化。换句话说，电势衡量的是单位正电荷在该位置所具有的电势能。
V = W/q
Here W is the work done, and q is the magnitude of the positive test charge. The SI unit of potential is the joule per coulomb (J C⁻¹), which is called the volt (V). One volt means one joule of work is done per coulomb of charge moved.
式中 W 为所做的功，q 为正检验电荷的电荷量。电势的国际单位是焦耳每库仑（J C⁻¹），称为伏特（V）。1 伏特表示移动 1 库仑电荷需要做 1 焦耳的功。
It is essential to remember that electric potential is a property of the field at a point, not a property of the test charge. The test charge is only a tool used to measure or define the potential.
必须牢记，电势是电场中某点的属性，而不是检验电荷的属性。检验电荷只是用来测量或定义电势的工具。
2. Potential Difference and Work Done | 电势差与做功
The potential difference ΔV between two points A and B is defined as the work done per unit positive charge when moving a positive test charge from A to B. It is often written as V(B) − V(A), where V(B) is the final potential and V(A) is the initial potential.
两点 A 与 B 之间的电势差 ΔV 定义为单位正电荷从 A 移动到 B 时所做的功。它通常写作 V(B) − V(A)，其中 V(B) 为末电势，V(A) 为初电势。
ΔV = V(B) − V(A) = W/q
For a positive charge moving from high potential to low potential, ΔV is negative, so the field does positive work and the charge gains kinetic energy if no other forces act.
若电荷 q 通过电势差 ΔV，电场对电荷做的功为 W = qΔV。对于从高电势移向低电势的正电荷，ΔV 为负值，因此电场做正功；若无其他力作用，电荷将获得动能。
For a negative charge, the situation is reversed: a negative charge accelerates from low potential to high potential because the electric force on it opposes the direction of increasing potential. Students must pay close attention to the sign of q and ΔV in numerical problems.
对于负电荷，情况正好相反：负电荷从低电势向高电势加速，因为作用在负电荷上的电场力与电势升高的方向相反。学生在计算题中必须特别注意 q 和 ΔV 的正负号。
3. Scalar Nature of Electric Potential | 电势的标量特性
Electric potential is a scalar quantity, unlike electric field strength, which is a vector. This means potential has magnitude but no direction at a given point. The scalar nature of potential greatly simplifies calculations when several charges are present, because potentials can be added algebraically without considering angles or vector components.
电势是标量，而电场强度是矢量。这意味着电势在某点只有大小而没有方向。电势的标量特性极大地简化了存在多个电荷时的计算，因为电势可以直接代数相加，无需考虑角度或矢量分量。
- Electric potential: scalar, unit V. / 电势：标量，单位 V。
- Electric field strength: vector, unit N C⁻¹ or V m⁻¹. / 电场强度：矢量，单位 N C⁻¹ 或 V m⁻¹。
Because potential is scalar, the sign of each contributing potential is simply positive or negative. A positive source charge creates a positive potential around it, while a negative source charge creates a negative potential. Zero potential is normally chosen at infinity for point charges.
由于电势是标量，各贡献电势的符号仅为正或负。正源电荷在其周围产生正电势，负源电荷产生负电势。对于点电荷，通常选择无穷远处为零电势点。
4. Electric Potential Due to a Point Charge | 点电荷产生的电势
For a point charge Q, the electric potential at a distance r from the charge is given by:
对于点电荷 Q，在距其 r 处的电势由下式给出：
V = Q / (4πε₀r)
. The potential is directly proportional to Q and inversely proportional to the distance r. For a positive point charge, V is positive and decreases to zero as r increases. For a negative point charge, V is negative and increases to zero as r increases.
式中 ε₀ 为真空介电常数，约等于 8.85 × 10⁻¹² F m⁻¹。电势与 Q 成正比，与距离 r 成反比。对于正点电荷，V 为正值，并随 r 增大而减小至零；对于负点电荷，V 为负值，并随 r 增大而增大至零。
The graph of V against r for a point charge is a hyperbola. Unlike electric field strength, which falls off as 1/r², potential falls off as 1/r. This difference is frequently tested in A-Level Physics.
点电荷的 V–r 图像为双曲线。与电场强度按 1/r² 衰减不同，电势按 1/r 衰减。这一区别是 A-Level 物理常考内容。
5. Superposition of Electric Potentials | 电势的叠加原理
Since electric potential is a scalar, the total potential at a point due to several point charges is simply the algebraic sum of the individual potentials. This is called the principle of superposition for potentials.
由于电势是标量，多个点电荷在某点产生的总电势就是各电荷单独存在时产生的电势的代数和。这称为电势的叠加原理。
V(total) = V₁ + V₂ + V₃ + ... = Σ Qᵢ / (4πε₀rᵢ)
For example, at the midpoint between two equal positive charges, the two potentials add to a large positive value, but the electric field at that same midpoint is zero because the field vectors cancel. This shows that total potential and total field can behave very differently.
例如，在两个相等正电荷的中点处，两个电势相加得到一个较大的正值，但该点的电场强度为零，因为电场矢量相互抵消。这表明总电势与总电场的行为可能截然不同。
Superposition also allows the potential to be zero at points where the electric field is not zero. For two equal but opposite charges, the potential is zero on the perpendicular bisector between them, yet the electric field is strong and directed across that plane. This is a classic exam scenario.
叠加原理也允许电势为零的位置电场不为零。对于两个等量异号电荷，其中垂面上电势为零，但电场却很强且方向穿过该平面。这是典型的考试情境。
6. Uniform Electric Fields and Potential Gradient | 均匀电场与电势梯度
In a uniform electric field, such as that between two parallel charged plates, the potential changes linearly with distance. The relationship between electric field strength E and potential difference ΔV is:
在均匀电场中，例如两块平行充电金属板之间的电场，电势随距离线性变化。电场强度 E 与电势差 ΔV 的关系为：
E = ΔV / d
at the electric field strength is equal to the potential gradient, meaning the rate at which potential changes with distance.
式中 d 为两板之间的垂直距离。该式也可写作 ΔV = Ed。它表明电场强度等于电势梯度，即电势随距离的变化率。
The unit of electric field strength is therefore equivalent to volts per metre (V m⁻¹), which is identical to newtons per coulomb (N C⁻¹). In a uniform field, a positive charge always feels a force from higher potential to lower potential, so the field direction is from the positive plate to the negative plate.
因此，电场强度的单位可等价表示为伏特每米（V m⁻¹），与牛顿每库仑（N C⁻¹）相同。在均匀电场中，正电荷受到的力总是从高电势指向低电势，因此电场方向由正极板指向负极板。
7. Relation Between Electric Field and Potential | 电场与电势的关系
More generally, the electric field is equal to the negative of the potential gradient. In one dimension this is written as:
更一般地，电场强度等于电势梯度的负值。在一维情况下可写作：
E = −dV/dr
The negative sign indicates that the electric field points in the direction in which the potential decreases most rapidly. A positive charge accelerates from high potential to low potential, while a negative charge accelerates from low potential to high potential.
负号表示电场方向指向电势下降最快的方向。正电荷从高电势向低电势加速，负电荷则从低电势向高电势加速。
r²). This confirms that the radial electric field is strongest near the charge and falls off rapidly with distance.
对于点电荷，将 V = Q/(4πε₀r) 对 r 求导，可得到熟悉的库仑电场 E = Q/(4πε₀r²)。这证实了点电荷的径向电场在电荷附近最强，并随距离迅速减弱。
ntial gradient is non-uniform, so E varies with position and must be found by differentiation or from the inverse-square law.
有用的对比：在均匀电场中，E = ΔV/d；在径向电场中，电势梯度不均匀，因此 E 随位置变化，必须通过微分或平方反比定律求得。
8. Equipotential Surfaces | 等势面
An equipotential surface is a surface on which every point has the same electric potential. Moving a charge along an equipotential surface requires no work because the potential difference is zero.
等势面是每个点电势都相同的面。沿等势面移动电荷不需要做功，因为电势差为零。
- For an isolated point charge, equipotential surfaces are concentric spheres centred on the charge. / 对于孤立点电荷，等势面是以电荷为中心的同心球面。
- For a uniform electric field, equipotential surfaces are planes perpendicular to the field lines. / 对于均匀电场，等势面是垂直于电场线的平面。
- Electric field lines always intersect equipotential surfaces at right angles. / 电场线总是与等势面垂直相交。
Equipotential surfaces are widely used to visualise fields. The closer the equipotential surfaces are to each other, the stronger the electric field, because the same potential change occurs over a smaller distance. This gives a higher potential gradient and therefore a larger magnitude of E.
等势面广泛用于直观展示电场。等势面越密集，电场越强，因为相同的电势变化发生在更短的距离内。这产生更大的电势梯度，因此 E 的大小也更大。
9. Electric Potential Energy of a System | 系统的电势能
The electric potential energy U of a charge q at a point where the electric potential is V is given by:
当电荷 q 位于电势为 V 的位置时，其电势能 U 为：
U = qV
For two point charges Q and q separated by a distance r, the electric potential energy of the pair is:
对于相距 r 的两个点电荷 Q 和 q，这对电荷的电势能为：
U = Qq / (4πε₀r)
The sign of U is important. If Q and q have opposite signs, U is negative, meaning work must be done to separate them to infinity; they are in a bound state. If the charges have the same sign, U is positive, and they naturally repel each other, releasing potential energy as kinetic energy when allowed to move apart.
U 的正负号很重要。如果 Q 与 q 异号，U 为负，表示将它们分离至无穷远需要外界做功；它们处于束缚状态。如果两电荷同号，U 为正，它们自然相互排斥，在允许分开时电势能转化为动能。
For a system of more than two charges, the total potential energy is the sum of the potential energies for every pair of charges. This pairwise summation avoids double counting and is a standard application of the scalar nature of potential energy.
对于两个以上电荷组成的系统，总电势能是每一对电荷的电势能之和。这种成对求和避免了重复计算，也是电势能标量特性的标准应用。
10. Electronvolt and Atomic Scales | 电子伏特与原子尺度
The electronvolt (eV) is a unit of energy defined as the kinetic energy gained by an electron when it is accelerated through a potential difference of exactly 1 volt. This unit is extremely convenient in atomic, nuclear and particle physics.
电子伏特（eV）是能量单位，定义为一个电子经过恰好 1 伏特电势差加速后所获得的动能。这个单位在原子、核与粒子物理中极为方便。
1 eV = 1.60 × 10⁻¹⁹ J
60 × 10⁻¹⁹ C) × (1 V) = 1.60 × 10⁻¹⁹ J. Therefore 1 eV is exactly this quantity of energy.
由于电子电荷 e = 1.60 × 10⁻¹⁹ C，将这一电荷移动通过 1 V 所做的功为 W = qΔV = (1.60 × 10⁻¹⁹ C) × (1 V) = 1.60 × 10⁻¹⁹ J。因此 1 eV 恰好等于这一能量。
drogen is 13.6 eV, which is much easier to compare than writing 2.18 × 10⁻¹⁸ J every time.
原子能级、电离能和光子能量常用电子伏特表示。例如，氢的电离能为 13.6 eV，这比每次都写 2.18 × 10⁻¹⁸ J 更容易比较。
11. Conductors in Electrostatic Equilibrium | 静电平衡中的导体
When a conductor reaches electrostatic equilibrium, the entire conductor becomes an equipotential volume. No potential difference can exist between any two points inside the conductor, because any such difference would drive a current until equilibrium is restored.
当导体达到静电平衡时，整个导体成为一个等势体。导体内部任意两点之间不存在电势差，因为任何电势差都会驱动电流，直到恢复平衡。
- The electric field inside a conductor is zero. / 导体内部电场为零。
- Any net charge resides entirely on the outer surface of the conductor. / 任何净电荷只分布在导体的外表面。
- The electric field just outside the surface is perpendicular to the surface. / 导体表面外侧的电场垂直于表面。
al is constant inside the conductor, its gradient is zero, so the electric field must also be zero. The surface of the conductor is therefore an equipotential surface, and field lines must meet it at right angles.
这些结论可直接从 E = −dV/dr 得出。如果导体内部电势恒定，则其梯度为零，因此电场也必为零。导体表面因而是一个等势面，电场线必须与其垂直相交。
12. Common Misconceptions and Exam Tips | 常见误区与考试提示
One common mistake is to assume that if the electric potential is zero at a point, the electric field must also be zero there. In fact, potential and field are different physical quantities, and a point can have zero potential but a strong electric field, as in the midpoint between two equal and opposite charges.
一个常见误区是认为若某点电势为零，则该点电场也必为零。事实上，电势与电场是不同的物理量，某点可以电势为零但电场很强，例如两个等量异号电荷的中点。
- Misconception: potential is a vector. Correction: potential is a scalar; only electric field is a vector. / 误区：电势是矢量。纠正：电势是标量，只有电场是矢量。
- Misconception: work depends on the path taken. Correction: the electrostatic field is conservative, so work done depends only on the initial and final potentials. / 误区：做功与路径有关。纠正：静电场是保守场，做功只取决于初末电势。
- an equipotential surface requires energy. Correction: no work is needed because ΔV = 0. / 误区：沿等势面移动电荷需要能量。纠正：因为 ΔV = 0，所以不需要做功。
- Misconception: negative potential means less than zero energy in an absolute sense. Correction: the zero of potential is chosen arbitrarily at infinity for point charges; only potential differences have physical significance. / 误区：负电势意味着能量绝对小于零。纠正：点电荷的电势零点是在无穷远处任意选择的；只有电势差才具有物理意义。
on asks for work done, check the sign carefully: work done by the field and work done against the field differ by a minus sign.
在考试答题时，定义电势时务必说明是将正检验电荷从无穷远处移来。计算点电荷电势时要包含 Q 的正负号，并在焦耳与电子伏特之间用 1 eV = 1.60 × 10⁻¹⁹ J 换算。当题目问到做的功时，要仔细检查正负号：电场做的功与克服电场做的功相差一个负号。
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