大学英文版电磁学讲义1-8

The Biot-Savart law is an empirical law(经验定律). It is a law for steady current. Permeability of vacuum(真空磁导率), 0=4 ×10−7 Tm/A . Example 1 Determine the magnetic field due to a current I in a long straight wire. (Fig, 8.8) , The field point x The direction of I is k in cylindrical coordinates is dz ' ( R , , z ), the line element d l is k z' . , the source point is x ' =k
v y t =−c 1 sin t c 2 cos t
(8.14) (8.15)
With the initial values v x 0 =v y 0 = 0 , the constants are c1 =− E / B , c2 = 0 Then v x t = E E 1 −cos t , v y t = sin t . B B
mv 2 =− F r = qvB R
(8.6)
The angular speed of the particle is
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Chapter 8 Magnetostatics 静磁学 v qB = = R m
(8.7)
is called cyclotron frequency(回旋频率). The period of revolution is 2 / .
Mass Spectrometer 质谱仪
A mass spectrometer is a device to separate charged particles by mass. In the velocity selector(速度选择器) there are crossed E and B fields, E= E j and B = B k . The particle with velocity v = E / B i can pass through the field undeflected. In the magnetic deflector(偏转器), the orbit of a particle is proportional to the ratio m/q, so different particle species travel on different circular path, separate in space.
The Biot-Savart law is an inverse-square law(平方反比定律). The direction of the 97
8.3 Electric Current as a Source of Magnetic Field magnetic field is azimuthal, around the axis of I d l . (Fig. 8.7) For a macroscopic wire, the magnetic field B(x) is B x = I d l× r 0 I d l× x − x ' 0 = ∫ ∫ 2 4 wire 4 wire r ∣x − x '∣3 (8.21)
d F = n L d l q v × B = I d l × B
(8.4) (8.5)
The total force on the wire is F =∫wire Id l ×B
8.2 Application of the magnetic force 磁力的应用 8.2.1 Helical or Circular Motion of q in Uniform B 运动电荷在均匀磁场中的螺旋或圆运动
. Suppose B = B k The equation of motion of the charge is dp =q v ×B dt F z = 0 => v z = constant . If v z = 0 , v is in x y plane and F ⊥ v , the charge is in uniform circular motion(匀 速圆周运动). If v z ≠ 0 , the motion is helical(螺旋的). the general trajectory is helix. In circular motion with radius R,
F =q v ×B
(8.1)
Equation (8.1) is the definition of magnetic field. (从测量的角度定义磁场) SI unit of magnetic field: tesla(特斯拉)(T): 1 newton per ampere-meter (C m/v). The force on 1 C moving 1 m/s perpendicular to a field of 1 T is 1 N. Magnetic force does no work on q. (磁场对电荷不做功)
(8.3)
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8.1 The Magnetic Force and the Magnetic Field
8.1.2 ForcБайду номын сангаас on a Current -Carrying Wire
作用在载流导线上的力
Electric current is motion of charges. Suppose the current consists of particles with charge q and linear density n L , moving with mean velocity v. The net force on a small segment d l of the wire is
dW =F⋅d x = F⋅v dt =0
(8.2)
The magnetic force affects the direction of motion of q, but not its kinetic energy(动能). Lorentz force(洛伦兹力) F =q E v × B
v D= E i B
(8.18)
The drift velocity does not depend on q. The positive and negative charges drift together.
8.2.3 Electric Motors 电动机
In a simple DC(直流电) motor the current is driven by a constant EMF(电动势). The current exists in a coil of wire, which is free to rotate on an axle. The magnetic force acts in opposite direction on opposite side of the coil, creating a torque(力矩) on the coil. To keep the direction of the torque constant as the coil rotates, the current in the coil must reverse every half cycle. The switch that reverses the current direction is called commutator(交换器).
8.2.2 Cycloidal Motion of q in Crossed E and B 电荷在垂直电磁场中的旋轮线运动
The motion of a positive charge q, that is released from rest(从静止释放) at the , is cycloidal. origin in the orthogonal uniform fields E= E j and B = B k
E t − sin t B
With the integral of v x t and v y t , the trajectory(轨迹) of q is
x t =
(8.16) (8.17)
y t =
E 1 −cos t B
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Chapter 8 Magnetostatics 静磁学 These are parametric equation(参数方程) for a cycloid curve(旋轮线). The motion of q is a combination of harmonic oscillations(简谐振动) in x and y, and a constant drift in the x direction. The drift velocity(飘移速度) is
Cyclotron 回旋加速器
A cyclotron is a charged particle accelerator(加速 器). Uniform B perpendicular to the orbit. The direction of E reverses for each half cycle. The frequency of oscillation(振动频率) of E in Hz is ω/(2π ). The cyclotron frequency remains constant as the particles gain energy, as long as v << c. For higher energies, i.e., v is a significant fraction of c, relativistic effects become important.
The solution of this inhomogeneous linear equation (非齐次线性方程)(8.12) is v x t = E c 1 cos t c2 sin t B (8.13)
Substitute (8.13) into (8.9), we get
Chapter 8 Magnetostatics 静磁学
Chapter 8 Magnetostatics 静磁学
8.1 The Magnetic Force and the Magnetic Field 磁力和磁场 8.1.1 Force on a Moving Charge 作用在运动电荷上的力
The force on a charge q moving with velocity v, exerted by a magnetic field B, is (磁场作用在以速度 v 运动的电荷上的力)
8.3 Electric Current as a Source of Magnetic Field 电流作为磁场的源 8.3.1 The Biot-Savart Law 毕奥-萨伐尔定律
Electric current is a source of magnetic field. 电流是磁场的源 The field dB at a point P, due to an infinitesimal current element I d l at a point P' is d B= r 0 I d l × 4 r2 (8.20)
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8.2 Application of the magnetic force
The equation of motion is m dv = q E v × B dt (8.8)
In component form d v x / dt = v y d v y / dt =− v x d v z / dt =0 => d 2 vx d t2 2 v x = 2 E B E q = (with ) B m B (8.9) (8.10) (8.11) (8.12)