Chapter 4. Maxwell’s Third Law
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1. Overview
2. Determination of Magnetic Field
4. Applications
a. Current Balancing Device and Self-Inductance Calculation
1. Overview
⑴ Maxwell’s Third Law
① When external current flows, magnetic force is generated.
② The intensity of the magnetic field (H) is determined by magnetic force and magnetic resistance.
⑵ Magnetic flux or magnetic flux density: Number of magnetic lines passing through an arbitrary cross-sectional area.
① Denoted by Φ.
② Unit: Weber (Wb)
⑶ Magnetization or magnetic flux density: Magnetization per unit area, i.e., B = Φ / A
① Denoted by B.
② Unit: Wb / m² = Tesla (T) = 10,000 Gauss (G)
⑷ Magnetic field
① Denoted by H.
② Terminology confusion
○ Generally, magnetic field refers to B.
○ Strictly, magnetic field means H.
○ B will be used to represent magnetic field from now on.
② Vacuum or air: Magnetic flux density B and magnetic field intensity H are linearly related.
③ Magnetic materials
○ Represented by hysteresis curve or B-H curve.
Figure 1. Hysteresis Curve
○ Retentivity
○ Coercivity
○ High Retentivity: Permanent magnets, advantageous for memory devices.
○ Low Retentivity: Advantageous for AC circuit magnetic circuits.
○ Area Dimension Analysis: ［B × H］= J / m³ = Energy per unit volume
○ Area under one cycle of B-H curve = Energy stored in the magnet and then lost as heat = Energy loss in AC operation.
○ Hysteresis loss
○ Measured by VSM (Vibrating Sample Magnetometer)
2. Determination of Magnetic Field
⑴ Ampere’s Law
① Law
② Vacuum or air
③ Right-hand screw rule: The direction in which the current flows is determined by the right-hand thumb, and the direction in which the other fingers wrap around is the direction.
④ Example 1. Magnetic field of infinitely long straight wire
○ When two infinitely long straight wires have the same direction of current flow
Figure 2. When two infinitely long straight wires have the same direction of current flow
○ When two infinitely long straight wires have opposite directions of current flow
Figure 3. When two infinitely long straight wires have opposite directions of current flow
⑤ Example 2. Solenoid
⑥ Example 3. Toroid: Solenoid bent into a donut shape
⑵ Biot-Savart Law
① Definition
② Application 1. Magnetic field at the center of a circular wire with radius R and current I
③ Application 2. Magnetic field at a point a distance r away from a circular loop of radius (b) carrying current (i)
④ Application 3. Magnetic field at a point slightly off the z-axis
Figure 4. Magnetic field at a point slightly off the z-axis
3. Lorentz Force
⑴ Definition: The force (magnetic force) experienced by a wire carrying charge or current under a magnetic field
⑵ Determination of the direction of magnetic force
① Fleming’s Left-hand Rule
○ If the current direction (+ → -) is the left hand thumb and the magnetic field direction (N → S) is the left hand index finger, the left hand thumb becomes the direction of Lorentz force.
○ Fleming’s Right-hand Rule is used in generators involving Maxwell’s Fourth Law.
Figure 5. Fleming’s Left-hand Rule
② Right-hand Rule of Magnetic Force
○ If the current direction (+ → -) is the right-hand thumb and the magnetic field direction (N → S) is the other fingers of the right hand, the palm becomes the direction of the force.
Figure 6. Right-hand Rule of Magnetic Force
③ Determination of direction using vector cross product (Preferred)
○ Using the right hand, grasp the current direction and magnetic field direction in order; the thumb points in the direction of Lorentz force.
Figure 7. Direction of cross product of vectors
⑶ Example 1. Force exerted on a wire carrying current: Current I, length L
⑷ Example 2. Force experienced by a moving charge within a constant magnetic field: Charge q, velocity v
⑸ Example 3. Electric motor
Figure 8. Principle of a Motor
Brushes, i.e., commutators, are clearly shown in the figure above.
① The direction of the rotational force changes periodically, causing oscillatory motion.
② Commutator
○ Device that ensures the coil of the electric motor rotates in a consistent direction.
○ Allows current to flow through the coil via contact with the brushes.
○ Brushes: Reverse the direction of current flow through the coil every half turn, ensuring continuous rotation in the same direction.
③ Electric motors have brushes, while generators do not due to alternating current (AC).
⑹ Example 4. Critical speed v for the path of the charge to remain straight
Figure 9. Problem Situation
⑺ Example 5. Hall Effect
Figure 10. Hall Effect
① The magnetic flux generated by the D-shaped wire and the conductor plate causes induced current to flow in the -y direction in the conductor plate.
② Case 1. When the charge carrier is positively charged
○ Moves in the -y direction due to positive charge.
○ Also moves in the -x direction due to Lorentz force.
○ Expected Result 1. Magnetic force is generated in the -x direction in the conductor plate.
○ Expected Result 2. The potential at point b is higher than at point a in the conductor plate.
③ Case 2. When the charge carrier is negatively charged
○ Moves in the +y direction due to negative charge.
○ Also moves in the -x direction due to Lorentz force.
○ Expected Result 1. Magnetic force is generated in the -x direction in the conductor plate.
○ Expected Result 2. The potential at point b is higher than at point a in the conductor plate.
④ Actual result
○ The potential at point b is higher than at point a.
○ Conclusion: The charge
carrier is negatively charged.
4. Applications
⑴ Electric motor
⑵ Speaker
⑶ Cyclotron
① Principle: Lorentz force = Centripetal force
② Purpose: Generation of radioactive isotopes, particle experiments
③ Bremsstrahlung: Light is generated by the accelerated motion of charged particles
⑷ Hard disk
⑸ Magnetic resonance imaging device
⑹ Electromagnetic balance scale
⑺ Maglev train
⑻ Tokamak
⑼ Magnetoencephalography (MEG)
⑽ Solenoid valve
⑾ Voice coil motor in a digital camera
Input: 2019-07-15 00:21
Modified: 2020-03-28 13:37