Chapter 27 Magnetic Field and Forces

Magnetic Interactions Fundamentals

Like Magnetic Poles Repel Each Other 同磁极相斥

When you bring two magnets close to each other and both have the same magnetic orientation (either both are north poles or both are south poles), they will push each other away. This is a manifestation of the principle that like magnetic poles repel. 当您将两个磁铁靠近，且两者具有相同的磁方向（即都是北极或都是南极），它们会互相推开。这是同磁极相斥原理的表现。

Opposite Magnetic Poles Attract 异磁极相吸

If you bring two magnets close and one has a north pole while the other has a south pole, they will be drawn toward each other. This is due to the principle that opposite magnetic poles attract. 如果您将两个磁铁靠近，一个具有北极而另一个具有南极，它们将互相吸引。这是因为异磁极相吸原理。

Earth's Magnetic Field 地球的磁场

Earth behaves as if it has a giant bar magnet inside it, with a north magnetic pole near the geographic South Pole, and a south magnetic pole near the geographic North Pole. This might seem counterintuitive, but it's the way the Earth's magnetic field lines up. 至于地球的磁场，它是一个复杂的现象。地球表现得好像它内部有一个巨大的磁铁，靠近地理南极有一个北磁极，靠近地理北极有一个南磁极。这可能看起来与直觉相违背，但这是地球磁场的排列方式

It's important to note that the Earth's magnetic field is not entirely aligned with its geographic axis, and it can vary in strength and orientation over time. This variation is essential for phenomena like compass navigation. 需要注意的是，地球的磁场并不完全与其地理轴线对齐，而且它的强度和方向随时间会发生变化。这种变化对于罗盘导航等现象至关重要。

Magnetic Polarization and Dipoles

Unique Properties 独特的特性

Magnetic materials exhibit unique properties when interacting with magnetic fields. Unlike electric polarization, magnetic materials can retain their magnetic polarization for extended periods. Understanding the behavior of magnetic materials, their interaction with magnetic fields, and the long-range behavior of magnetic fields is essential for various applications.

磁性材料在与磁场相互作用时表现出独特的特性。与电极化不同，磁性材料可以长时间保持其磁极化状态。理解磁性材料的行为、它们与磁场的相互作用以及磁场的长程行为对于各种应用至关重要。

Magnetic Materials Attracted by Both Magnetic Poles 磁性材料被两种磁极吸引

Magnetic materials are attracted to both types of magnetic poles, whether it's the north or south pole of a magnet. This is due to the nature of magnetic forces, which differ from electric forces. Unlike electric charges, which can be either positive or negative, magnetic poles always come in pairs, making the attraction to both poles a characteristic feature.

磁性材料会被两种磁极吸引，无论是磁铁的北极还是南极。这是由于磁力的性质，不同于电力，磁力总是成对出现，使被两种磁极吸引成为一个特征。

Magnetic Polarization Induced by Magnetic Fields 磁场诱导磁性极化

The presence of magnets or magnetic fields can induce magnetic polarization in nearby magnetic materials. This is similar to electric polarization, where the atomic or molecular magnetic moments align in response to the external magnetic field. 磁铁或磁场的存在可以在附近的磁性材料中诱导磁性极化。这类似于电极化，其中原子或分子的磁矩会在外部磁场的作用下排列。

Retention of Magnetic Polarization 磁性极化的保持

One notable distinction from electric polarization is that magnetic materials can retain their magnetic polarization for extended periods, even after the external magnetic field is removed. This makes them useful in a variety of applications, including magnetic data storage and permanent magnets. 与电极化的一大区别是，磁性材料可以在外部磁场移除后保持其磁性极化状态，甚至保持很长时间。这使它们在各种应用中非常有用，包括磁性数据存储和永久磁铁。

Long-Range Behavior of Magnetic Fields 磁场的长程行为

The long-range behavior of magnetic fields can be thought of in a manner similar to gravitational and electric fields. The behavior depends on the principle of superposition, which states that the total magnetic field at a point is the vector sum of the fields produced by all magnetic sources in the vicinity. Additionally, the behavior depends on the permeability of the material through which the magnetic field propagates. Materials can be classified into paramagnetic, diamagnetic, and ferromagnetic based on their responses to external magnetic fields. 磁场的长程行为可以类似于重力和电场。其行为依赖于叠加原理，该原理规定了某一点的总磁场是附近所有磁源产生的磁场的矢量和。此外，行为还取决于磁场传播的材料的磁导率。根据对外部磁场的响应，材料可以分为顺磁性、抗磁性和铁磁性。

Magnetic Field Lines

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Closed Loops that Extend from the North Pole to the South Pole 磁场线形成封闭的回路，从磁铁的北极延伸到南极。 field lines are closer together, the magnetic field is stronger 磁场线越密集的地方，磁场就越强 unit: tesla [T]

Magnetic Field Lines Representation 磁场线的表示

A magnet is surrounded by a magnetic field, and these field lines provide a convenient way to illustrate the spatial distribution and direction of the magnetic field. At any point, the direction of the magnetic field is tangent to the magnetic field line passing through that position. This visualization aids in understanding the orientation and strength of the magnetic field at different locations around the magnet.

磁铁周围存在磁场，而这些磁场线为以图形方式展示磁场的空间分布和方向提供了便捷的方法。在任何点上，磁场的方向都是切线于通过该位置的磁场线。这种可视化有助于理解磁场在磁铁周围不同位置的方向和强度。

Formation of Magnetic Field Lines 磁场线的形成

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closed loops that extend from the north pole to the south pole

Magnetic field lines form closed loops that extend from the north pole to the south pole of a magnet. They exit the magnet at the north pole and re-enter at the south pole, creating a continuous and closed pathway for the magnetic field.

磁场线形成封闭的回路，从磁铁的北极延伸到南极。它们从北极退出，再从南极重新进入，形成了磁场的连续闭合路径。

Relationship between Magnetic Field and Field Line Density 磁场与磁场线密度的关系

The magnitude of the magnetic field at any location is directly proportional to the density of field lines at that location. This implies that where field lines are closer together, the magnetic field is stronger, and where they are more spaced out, the field is weaker. 在任何位置，磁场的大小与该位置的磁场线密度成正比。这意味着磁场线越密集的地方，磁场就越强，而它们分散的地方，磁场就越弱。

Unit of Magnetic Field 磁场的单位

The magnetic field is quantified in the International System of Units (SI) using the unit tesla [T]. One tesla is equivalent to one newton per ampere-meter, emphasizing the force exerted on a charged particle moving perpendicular to the magnetic field.

磁场在国际单位制（SI）中以特斯拉[T]为单位进行量化。一特斯拉等于每安培米的牛顿，强调了对垂直于磁场移动的带电粒子施加的力。

Magnetic Field Flux

Similarities with Electric Field Flux 与电场通量的相似性

The concept of magnetic flux shares fundamental principles with electric field flux. Both represent the flow of a field through a surface and are integral to understanding the interactions between fields and surfaces.

磁通量的概念与电场通量共享基本原理。两者都表示场通过表面的流动，对于理解场与表面之间的相互作用至关重要。

\Phi_{B} \equiv B A \cos(\theta) =\vec{B} \cdot \vec{A}

Scalar Nature and SI Units 标量性质和SI单位

Magnetic flux is a scalar quantity, representing the total magnetic field passing through a surface. Its SI units are tesla·meter² (T·m²), and this unit is named the weber (Wb). The weber serves as a measure of the total magnetic field passing through a given surface.

磁通量是一个标量量，表示通过一个表面的总磁场。其国际单位为特斯拉·米²（T·m²），这个单位被命名为韦伯（Wb）。韦伯用于衡量通过给定表面的总磁场。

1 Wb \equiv 1 T\cdot m^2 = 1 m^2 \cdot kg/(s^2 \cdot A)

Non-Uniform Fields and Irregular Surfaces 非均匀场和不规则表面

When dealing with non-uniform magnetic fields or irregular surfaces, calculating magnetic flux involves integrating the magnetic field strength over the given surface. The formula for magnetic flux ( $Φ$ ) is analogous to the electric field flux ( $ΦE$ ) and is expressed as the dot product of the magnetic field vector ( $B$ ) and the surface area vector (A): $Φ = B · A$ . If the magnetic field or the surface varies, integration becomes necessary to account for these variations.

在处理非均匀磁场或不规则表面时，计算磁通量涉及对给定表面上的磁场强度进行积分。磁通量的公式（Φ）类似于电场通量（ΦE），表示为磁场矢量（B）和表面积矢量（A）的点积： $Φ = B · A$ 。如果磁场或表面发生变化，就需要进行积分以考虑这些变化。

\Phi_B = \int{\vec{B}\cdot d{\vec{A}}}

Gauss’s Law for Magnetism

The total magnetic flux through any closed surface equals zero. 通过任何封闭表面的净磁通量等于零.

\Phi_B = \int{\vec{B}\cdot d{\vec{A}}} = 0

Key Concepts of Gauss's Law for Magnetism

Gauss's Law for Magnetism states that the net magnetic flux through any closed surface is equal to zero. This means that there are no magnetic monopoles, and the magnetic field lines form continuous closed loops. Unlike electric field lines, which can originate or terminate at electric charges, magnetic field lines neither start nor end but always form complete loops. 高斯磁场定律指出，通过任何封闭表面的净磁通量等于零。这意味着不存在磁单极子，磁场线形成连续的封闭回路。与电场线不同，电场线可以起源或终止于电荷，磁场线既不开始也不结束，而是始终形成完整的回路。

Implications

Absence of Magnetic Monopoles: Gauss's Law for Magnetism implies the nonexistence of isolated magnetic charges, commonly known as magnetic monopoles. All magnetic field lines either enter or exit a magnet, forming closed loops. 磁单极子的不存在： 高斯磁场定律暗示了孤立的磁荷，通常称为磁单极子，是不存在的。所有的磁场线都要么进入磁铁，要么从磁铁出来，形成封闭回路。

Conservation of Magnetic Flux: The law underscores the conservation of magnetic flux. No magnetic flux is created or destroyed; it always forms complete loops. 磁通量的守恒： 该定律强调了磁通量的守恒。磁通量不会被创造或破坏，它总是形成完整的回路。

Symmetry in Magnetic Fields: The law is particularly useful when magnetic fields exhibit symmetry. The calculation of magnetic flux becomes more straightforward by choosing appropriate closed surfaces. 磁场的对称性： 当磁场呈现对称性时，这个定律特别有用。通过选择适当的封闭表面，磁通量的计算变得更为简单。

Magnetic Field Effect on Moving Charges

When a charged particle moves through a magnetic field, it experiences a force perpendicular to both its velocity vector (V) and the magnetic field vector. This force is known as the magnetic Lorentz force and can be determined using the Right Hand Rule.

当电荷粒子穿过磁场时，它会受到一个与其速度矢量（V）和磁场矢量都垂直的力，这被称为磁洛伦兹力。这个力可以通过使用右手法则来确定。

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Positively Charged Particle is the direction determined by Right Hand Rule Negatively Charged Particle is the opposite direction determined by Right Hand Rule

The Right Hand Rule 右手法则

Thumb: Point your thumb in the direction of the charged particle's velocity (V). 大拇指： 用大拇指指向带电粒子的速度方向（V）。

Index Finger: Point your index finger in the direction of the magnetic field (B). 食指：用食指指向磁场的方向（B）。

Middle Finger: The middle finger, perpendicular to both the thumb and index finger, represents the direction of the magnetic force (F) acting on the charged particle. 中指：中指垂直于大拇指和食指，表示作用在带电粒子上的磁力（F）方向。

Equations

Magnetic Force Equation

F^B_p= |q|vB\sin(\theta)

Vector Form of Magnetic Force

\vec{F^B_p}=q\vec{v}\times\vec{B}

Combined Electric and Magnetic Force

\vec{F^{EB}_p}=q(\vec{E}+\vec{v}\times\vec{B})

Example

Cyclotrons

Mass Spectrometers 质谱仪

Equation

|q|vB=\frac{mv^2}{R}

|q| is the magnitude of the charge, v is the velocity of the particle, B is the magnetic field strength, m is the mass of the particle, and R is the radius of the circular path the particle follows. |q|是电荷的大小，v是粒子的速度，B是磁场强度，m是粒子的质量，R是粒子所沿路径的圆形半径。

Mass Spectrometers:

Mass spectrometers are analytical instruments designed to separate and identify ions based on their mass-to-charge ratio (m/z). Charged particles (ions) generated from a sample are accelerated into a magnetic field. The force from the magnetic field causes these ions to move in circular paths. By applying the centripetal force equation, the mass-to-charge ratio of ions can be determined, allowing for precise identification and quantification of substances./

Magnetic Field Line & Current Carrying Wires

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Right Hand Rule: curled fingers the direction of the magnetic field lines

Magnetic Field Lines Around Current-Carrying Wires 载流导线周围的磁场线

When an electric current flows through a conductor, such as a wire, it generates a magnetic field around the wire. The direction of the magnetic field lines can be determined using the right-hand rule: if the thumb points in the direction of the current, the curled fingers indicate the direction of the magnetic field lines around the wire. 当电流通过导体（如导线）流动时，它在导线周围产生磁场。可以使用右手法则确定磁场线的方向：如果拇指指向电流的方向，弯曲的手指表示围绕导线的磁场线的方向。

Circular Pattern of Magnetic Field Lines 磁场线的圆形图案

The magnetic field lines around a straight current-carrying wire form concentric circles centered on the wire. The spacing between these circles represents the strength of the magnetic field – the closer the circles, the stronger the field.

直导线周围的磁场线形成以导线为中心的同心圆。这些圆之间的间距表示磁场的强度，圆越接近，磁场就越强。

Magnitude of the Magnetic Field 磁场的大小

The magnitude of the magnetic field (B) generated by a current-carrying wire is directly proportional to the current (I) passing through the wire and inversely proportional to the distance (r) from the wire. The mathematical expression is given by the formula: $B = \frac{(μ₀ * I) }{ (2π * r)}$ , where $μ₀$ is the permeability of free space. 由载流导线产生的磁场的大小（B）与通过导线的电流（I）成正比，与距离（r）的平方成反比。数学表达式由公式给出：B = (μ₀ * I) / (2π * r)，其中μ₀是自由空间的磁导率

Applications of Magnetic Fields from Current-Carrying Wires 载流导线磁场的应用

Understanding the magnetic field lines around current-carrying wires is crucial in various applications. It forms the basis for the operation of devices like solenoids, electromagnets, and transformers. Additionally, this knowledge is integral in designing efficient electrical systems and plays a role in fields such as power generation, transportation, and telecommunications. 理解载流导线周围的磁场线对于各种应用至关重要。它构成了诸如螺线管、电磁铁和变压器等设备的运行基础。此外，这一知识对于设计高效的电气系统至关重要，并在发电、交通和电信等领域发挥着作用。

Force on Current Carrying Wires by Magnetic Field

Magnetic Field and Current-Carrying Wires 磁场与载流导线

The impact of a magnetic field on a current-carrying wire is analogous to its effect on a moving charge. This correspondence makes sense considering that electric current can be conceptualized as the time rate of moving charged particles crossing a unit area of a conductor.

磁场对载流导线的影响类似于其对运动电荷的影响。考虑到电流可以被视为单位导体面积上移动带电粒子的时间速率，这种对应是合理的。

Force on Current-Carrying Wires in a Magnetic Field 磁场对载流导线的力

When a current-carrying wire is placed in a uniform external magnetic field, the magnetic field induces a force on the wire. This force can be experimentally demonstrated and is determined by the right-hand rule. According to the rule, if the thumb points in the direction of the current, the forefinger in the direction of the external magnetic field, the middle finger in the direction of the force, then the palm gives the direction a positive charge would move.

This phenomenon is a manifestation of the Lorentz force experienced by charged particles moving in a magnetic field. The force on the current-carrying wire is perpendicular to both the direction of the current and the external magnetic field.

当载流导线置于均匀外部磁场中时，磁场会在导线上诱导出一种力。这种力可以通过右手法则在实验中得到验证。根据该法则，如果拇指指向电流的方向，食指指向外部磁场的方向，中指指向力的方向，那么手掌方向表示正电荷将移动的方向。

这一现象是在磁场中运动的带电粒子所经历的洛伦兹力的体现。作用在载流导线上的力垂直于电流方向和外部磁场方向。

Magnetic Force on a Straight Wire Segment Ampère's force | 安培力

\vec{F} = I \vec{l}\times\vec{B}

$\vec F$ the Ampère's force, measured in Newtons (N). 表示安培力，单位是牛顿（N）。

$I$ current intensity, measured in Amperes (A). 是电流强度，单位是安培（A）。

$\vec{l}$ length vector of the current element, representing a small segment along the direction of the current. 是电流元的长度矢量，表示电流方向上的微小线段。

$\vec{B}$  the magnetic induction (magnetic field strength) vector, measured in Tesla (T). 是磁感应强度（磁场强度）矢量，单位是特斯拉（T）。

Magnetic Dipole Moment

\mu=I\cdot A

The magnetic dipole moment is a vector used to describe the direction and magnitude of a magnetic dipole. It points in the direction of the magnetic field at the center of the dipole. The magnitude of the magnetic dipole moment ( $\mu$ ) is given by the product of the current ( $I$ ) and the area ( $A$ ) of the loop:

磁偶极矩是一个矢量，用来描述磁偶极的方向和大小。它指向磁偶极在其中心处的磁场方向。磁偶极矩的大小与电流携带环的面积和电流强度有关，可以用公式表示为磁偶极矩（ $\mu$ ）等于电流（ $I$ ）乘以环的面积（ $A$ ）

It points in the direction of the magnetic field at the center of the dipole.

Forces on Current-Carrying Loops 电流携带环上的力

When a current flows through a closed loop, the loop experiences a torque due to the magnetic field. According to the principles of Lorentz force, this torque can be calculated using the formula: 当电流通过一个闭合回路时，该回路会受到磁场的作用，产生一个力矩。根据洛伦兹力的原理，这个力矩可以通过以下公式计算：

$τ=μ×B$ 

Here, $τ$  is the torque, $μ$  is the magnetic dipole moment, and $B$  is the magnetic field. This implies that the current-carrying loop undergoes a rotational torque in the magnetic field, aligning itself with the field direction. 其中， $τ$  是力矩， $μ$ 是磁偶极矩, $B$ 是磁场。这意味着电流携带环在磁场中会受到一个旋转力矩，导致它倾向于对齐到磁场方向。