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The net current flowing through the surface of the contour is zero because an equal number of current coils cross the surface in both direction.. Luong Vinh Quoc Danh 32 Applic[r]

Bài giảng: TRƯỜNG ĐIỆN TỪ (CT361)

(ELECTROMAGNETICS)

Chapter 5: Magnetostatics

(Từ tĩnh học)

• Magnetic Forces and Torques • The Biot-Savart Law

• Magnetic Forces Between Two Parallel Conductors • Maxwell’s Magnetostatic Equations

• Ampere’s Law

• Magnetic Boundary Conditions

Content

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Magnetostatics

Maxwell’s Equations: If we take out the time

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Force and Torque

Magnetic force:

Electromagnetic (Lorentz) force:

| Fm| = Fm = q.u.B.sin

+ Fe in the direction of E; Fm perpendicular to B.

Force on a Current-Carrying Conductor

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Moving charges  Current

Differential force dFm

on a differential current I dl:

(5.9)

Total force on a segment of current:

l

I d

  

m

F l B

For closed contour C in a uniform B field:

Example 5-1: Force on a semicircular conductor

(a) Determine F1 on the straight section of the wire and (b) the force F2 on the curved section

Application of Equ (5.12) with l = x2r

gives:

B = yB0

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Magnetic Torque on a Current-Carrying Loop

T = torque, d = moment arm, F = force Torque (momen xoắn)

Magnetic torque on current loop

Magnetic torque on current loop (cont.)

No forces on arms and (I and B are parallel, or anti-parallel)

Magnetic torque:

Inclined Loop

For a loop with N turns and whose

surface normal is at angle θ relative to

B direction:

Magnetic Moment

(5.18)

(5.21)

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Biot-Savart Law

Magnetic field induced by a differential current:

Magnetic Field due to Current Densities

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Example 5-2:

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Example 5-3

dH is in the r–z plane , and therefore it has

components dHr and dHz

z-components of the magnetic fields due to dl and dl’ add because they are in the same

direction, but their r-components cancel

Hence for element dl:

Magnitude of field due to dl is

Example 5-3 (cont.)

Magnetic Dipole

Because a circular loop exhibits a magnetic field pattern

similar to the electric field of an electric dipole, it is

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Forces on Parallel Conductors

Parallel wires attract if their currents are in the same direction,

Demos

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Maxwell’s Magnetostatic Equations

 Gauss’s Law for Magnetism

(5.47)

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Example 5-5: Internal Magnetic Field

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For r > a

Magnetic Field of Toroid

Applying Ampere’s law over contour C

The magnetic field

Example 5-6 (cont.)

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