Electric Potential | 电势

📚 Electric Potential | 电势

Electric potential is a scalar field that simplifies the analysis of electrostatic systems by relating to potential energy. In AP Physics C: Electricity and Magnetism, you will need to compute potentials, understand its gradient relation to the electric field, and apply it to conductors and charge distributions.

电势是一个标量场，通过与势能的联系简化了静电系统的分析。在AP物理C电磁学中，需要计算电势，理解它与电场的梯度关系，并将其应用于导体和电荷分布。

1. Introduction to Electric Potential | 电势简介

Electric potential (V) is defined as the electric potential energy (U) per unit charge (q). It is a scalar quantity measured in volts (1 V = 1 J/C). This concept allows us to describe the effect of an electric field without dealing with vectors, making many problems easier.

电势（V）定义为单位电荷（q）的电势能（U）。它是一个标量，单位是伏特（1 V = 1 J/C）。这一概念使我们无需处理矢量就能描述电场的效应，使许多问题变得简单。

V = U / q

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2. Definition and Formula | 定义与公式

The electric potential at a distance r from a point charge Q (taking zero at infinity) is given by V = kQ/r, where k = 1/(4πε₀) = 8.99×10⁹ N·m²/C². If the source charge is positive, the potential is positive; if negative, the potential is negative. This scalar nature simplifies superposition.

距离点电荷 Q 为 r 处的电势（取无穷远处为零）为 V = kQ/r，其中 k = 1/(4πε₀) = 8.99×10⁹ N·m²/C²。若源电荷为正，电势为正；若为负，则为负。这种标量特性简化了叠加计算。

V = kQ / r = (1 / (4π ε₀)) (Q / r)

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3. Potential Difference and Work | 电势差与功

The potential difference ΔV between two points A and B is the negative of the work done by the electric field per unit charge moving from A to B: ΔV = V_B − V_A = −Wₑ / q. The work done by an external force against the field is W_ext = q ΔV. For a charge moving freely, the change in kinetic energy is ΔK = −q ΔV.

两点 A 和 B 之间的电势差 ΔV 等于电场力对单位电荷从 A 移到 B 所做功的负值：ΔV = V_B − V_A = −Wₑ / q。外力反抗电场力做的功为 W_ext = q ΔV。对于自由运动的电荷，动能变化 ΔK = −q ΔV。

ΔV = −Wₑ / q

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4. Electric Potential of a Point Charge | 点电荷的电势

For a point charge, equipotential surfaces are concentric spheres. The potential decreases as 1/r. Inside a conductor in equilibrium, the potential is constant, equal to that on the surface. This property is used to protect sensitive equipment (Faraday cage).

点电荷的等势面是同心的球面。电势随着 1/r 减小。在平衡导体内部，电势恒定，等于表面的电势。这一特性可用于保护灵敏设备（法拉第笼）。

V ∝ 1 / r

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5. Superposition Principle | 叠加原理

The total potential at a point due to several charges is the algebraic sum of the individual potentials: V_total = V₁ + V₂ + … = Σ kQᵢ/rᵢ. Because potential is a scalar, you just add numbers with their signs, unlike vector addition for electric field. This makes calculating potentials for discrete charge distributions straightforward.

多个电荷在某点的总电势是各个电荷单独产生的电势的代数和：V_total = V₁ + V₂ + … = Σ kQᵢ/rᵢ。因为电势是标量，只需将带符号的数值相加，这与电场的矢量加法不同。这使离散电荷分布的电势计算变得直观。

V_total = Σᵢ (k Qᵢ / rᵢ)

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6. Relationship between Electric Field and Potential | 电场与电势的关系

The electric field is the negative gradient of the potential. In one dimension, E = −dV/dx. For a uniform field, the magnitude is E = |ΔV/Δd|. Field lines point from higher to lower potential. The potential variation can be used to find the field: if V(x) = 5x², then E(x) = −dV/dx = −10x. This relationship is fundamental for understanding how charges move.

电场是电势的负梯度。在一维情况下，E = −dV/dx。对于匀强电场，大小为 E = |ΔV/Δd|。电场线由高电势指向低电势。电势变化可用于求电场：若 V(x) = 5x²，则 E(x) = −dV/dx = −10x。这一关系对于理解电荷运动至关重要。

E = − dV / dx

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7. Equipotential Surfaces | 等势面

Equipotential surfaces are everywhere perpendicular to electric field lines. No work is done when moving a charge along an equipotential. In a uniform field, equipotentials are parallel planes. For a point charge, they are spheres. Conductors form equipotential volumes, which is why their surface is always an equipotential.

等势面处处与电场线垂直。沿着等势面移动电荷不做功。在匀强场中，等势面是平行平面；点电荷的等势面是球面。导体构成等势体，因此其表面始终是一个等势面。

W = 0 along an equipotential

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8. Potential Energy of a System of Charges | 电荷系统的电势能

The electric potential energy of two point charges q₁ and q₂ separated by r is U = k q₁ q₂ / r. For a system of many charges, the total potential energy is (1/2) Σᵢ Σⱼ,ⱼ≠ᵢ k qᵢ qⱼ / rᵢⱼ, where the factor 1/2 corrects double-counting. A test charge q₀ in a potential V has U = q₀ V. This concept is key in particle accelerators and electron orbits.

两个相距为 r 的点电荷 q₁ 和 q

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