Maxwell's Equations and Electromagnetic Induction, Exercises of Differential Equations

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Textbook : Field and Wave Electromagnetics

David K. Cheng

Time-Varying Fields and Maxwell’s

Equations

- Part I-

• Table

• In the electrostatic model
  • E and D are not related to B and H in the

magnetostatic model.

• In static conditions,
  • electric and magnetic fields are independent of

each other

• In a conducting medium,
  • static electric and magnetic fields may both exist.
• In the time varying case
  • Time-varying B can produce E
  • Time-varying D can produce H

• Static models are
  • simple,
  • but they are not suitable for explaining time- varying

electromagnetic phenomena.

• Static electric and magnetic fields do not produce waves

that propagate and carry energy and information.

• Waves are the essence of electromagnetic action at a

distance. Here, a changing magnetic field induces an

electric field, and vice versa.

• Under time-varying conditions the electric field vectors E

and D are properly related to the magnetic field vectors B

and H.

• Consider the expression again 𝛻 × 𝐄 = −

• Taking the surface integral of both sides over an open surface and applying Stoke’s theorem, we obtain (7-2)
• S is a surface bounded by the closed line C. This equation is valid for any surface S with a bounding contour C.
• In a field with no time variation, we have 𝜕𝐁 𝜕𝑡 = 0 , and above equations reduce to the static case equations.

Stationary Circuit in a Time-Varying Magnetic Field

• For a stationary circuit with a contour C and surface S , the above equation can be written as ර 𝐶

𝑆

• If we define
• Then we obtain

Example 1.

• A circular loop of N turns of conducting wire lies in the xy - plane with its center at the origin of a magnetic field specified by

B = a z B 0 cos(  r /2 b )sin  t ,

• where b is the radius of the loop and  is the angular frequency. Find the emf induced in the loop.

• The magnetic flux linking each turn of the circular loop is
• Since there are N turns, the total flux linkage is N . Then
• The induced emf is 90 

out of time phase with the

magnetic flux.

A Moving Circuit in a Time-Varying Magnetic Field

• When a charge q moves with a velocity u in a region where both an electric field E and magnetic field B exist, the electromagnetic force F on q is given by Lorentz's force equation : F = q ( E + u  B ),
• To an observer moving with q, there is no apparent motion, and the force on q can be interpreted as caused by an electric field E' , where E = E + u  B
• Then, when a conducting circuit with contour C and surface S moves with a velocity u in a field ( E , B ), we obtain
• This is the general form of Faraday’s Law for a moving circuit in a time- varying magnetic field. - The line integral on the left side is the emf induced in the moving frame of reference. - The first term on the right side represents the transformer emf due to the time variation of B. - The second term represents the motional emf due to the motion of the circuit in B.

Maxwell’s Equations

• The fundamental postulate for electromagnetic induction :
  • a time-varying magnetic field causes an electric field.
• Then revised set of two curl and two divergence equations are
• In addition, the principle of conservation of charge must be satisfied at all times.
• The expression for charge conservation is the equation of continuity :
• The previous equations should be consistent with above equation in a time- varying situation. However, Faraday’s Law Ampere’s Circuital Law Gauss’s Law No Isolated Magnetic Charge

• Differential form of Maxwell’s equations for time-varying fields :
•  is the volume charge density,
• J is the current density, (it may contain both convection current (  u ) and conduction current ( E ).
• These four equations, together with the equation of continuity and Lorentz' force equation form the foundation of electromagnetic theory. Maxwell’s Equations

Integral Form of Maxwell’s Equations

• Differential form of Maxwell's equations are valid at every point in space.
• To explain electromagnetic phenomena in physical environment we deal with finite objects of specified shapes and boundaries. It is convenient to convert the differential forms into their integral-form equivalents.
• Hence, we take the surface integral of both sides of the curl equations over an open surface S with a contour C and apply Stokes's theorem. Then we obtain
• Next, we take the volume integral of both sides of the divergence equations over a volume V with a closed surface S and use Divergence Theorem. Then we obtain
• These four equations are the integral form of Maxwell's equations.