Stationary Loop in a Time-Varying Magnetic Field ~ Displacement Current
KOCW – 전자기학(전자정보공학과 오창현 교수)
17, 18강 강의자료
Chapter 6.
Maxwell’s Equations for
Time-varying Fields
Faraday’s Law
• Oersterd, who demonstrated that a wire carrying an electric current exerts a force on a compass needle and that the needle always turns so as to point in the Φ direction when the current is along the Ƹ𝑧 direction.
• Faraday hypothesized that if a current produces a magnetic field, then the converse should also be true: a magnetic field should produce a current in a wire.
Magnetic fields can produce an electric current in a closed loop, but only if the magnetic flux linking the surface area of the loop changes with time. The key to the induction process is change.
Faraday’s Law
Faraday’s Law
• A galvanometer is a predecessor of the voltmeter and ammeter.
• This voltage is called the electromotive force(emf), 𝑉𝑒𝑚𝑓, and the process is called electromagnetic induction.
• An emf can be generated in a closed conducting loop under any of the following three conditions:
1. A time-varying magnetic field linking a stationary loop; the induced emf is then called the transformer emf, 𝑉𝑒𝑚𝑓𝑡𝑟 .
2. A moving loop with a time-varying area(relative to the normal component of B) in a static field B; the induced emf is then called the motional emf, 𝑉𝑒𝑚𝑓𝑚 3. A moving loop in a time-varying field B.
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Stationary Loop in a Time-Varying Magnetic Field
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The connection between the direction of ds and polarity of 𝑉𝑒𝑚𝑓𝑡𝑟 is governed by the following the right-hand, then the direction of the contour C indicated by the four fingers is such that it always passes across the opening form the positive terminal of 𝑉𝑒𝑚𝑓𝑡𝑟 to negative terminal
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The polarity of 𝑉𝑒𝑚𝑓𝑡𝑟 and hence the direction of I is governed by Lenz’s law, which states that the current in the loop is always in a direction that opposes the charge of magnetic flux Φ 𝑡 that produced I.
If it is important to remember that 𝐁𝑖𝑛𝑑 serves to oppose that charge in B(t), and not necessarily B(t) itself.
Stationary Loop in a Time-Varying Magnetic Field
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• Even though the derivation leading to Faraday’s law started out by considering the field associated with a physical circuit, Eq.(6.13) applies at any point in space.
Stationary Loop in a Time-Varying Magnetic Field
The Ideal Transformer
• In an ideal transformer the core has infinite permeability 𝜇 = ∞ , and the magnetic flux is confined within the core.
The directions of currents flowing in the two coils, 𝐼1 and 𝐼2, are defined such that, when 𝐼1 and 𝐼2 are both positive, the flux generated by 𝐼2 is opposite to that generated by 𝐼1. The transformer gets its name form the fact that it transforms currents, voltages, and impedances between its primary and secondary circuits, and vice versa.
The Ideal Transformer
• The flux Φ and voltage 𝑉1 are related by Faraday’s law:
• A similar relation holds true on the secondary side:
The Ideal Transformer
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Moving Conductor in a Static Magnetic Field
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Only those segments of the circuit that cross magnetic field lines contribute to 𝑉𝑒𝑚𝑓𝑚 .
Moving Conductor in a Static Magnetic Field
The Electromagnetic Generator
• A permanent magnet is used to produce a static magnetic field B in the slot between its two poles.
• The current flows in opposite directions in segments 1-2 and 3-4 of the loop.
• The loop is made to rotate by an external force.
• 𝑉𝑒𝑚𝑓𝑚 , as shown in Fig. 6-11(b), is being converted into electrical energy.
The Electromagnetic Generator
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• Segments 1-2 and 3-4 of the loop are of length l
• Neither crosses the B lines when the loop rotates.
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The Electromagnetic Generator
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The Electromagnetic Generator
• This same result can also be obtained by applying the general form of Faraday’s law given by Eq.(6.6). The flux linking the surface of the loop is
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The Electromagnetic Generator
The voltage induced by the rotating loop is sinusoidal in time with an angular frequency 𝜔 equal to that of the rotating loop, and its amplitude is equal to the product of the surface area of the loop, the magnitude of the magnetic field generated by the magnet, and the angular frequency 𝜔.
Moving Conductor in a Time-Varying Magnetic Field
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Displacement Current
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Displacement Current
• The second term on the right-hand side of Eq.(6.43) of course has the same unit (amperes) as the current 𝐼d, and because it is proportional to the time derivative of the electric flux density D, which is also called the electric displacement, it is called the displacement current 𝐼d. That is,
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where 𝐉d = 𝜕𝐃/𝜕𝑡 represents a displacement current density.
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Displacement Current
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Displacement Current
• In the perfect dielectric with permittivity 𝜖 between the capacitor plates, 𝜎 = 0. Hence, 𝐼2𝑐 = 0 because no conduction current exist there.
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Even though the displacement current does not transport free charges, it nonetheless behaves like a real current.
