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Worked Examples · Example 6.5

Q.(a) A closed loop is held stationary in the magnetic field between the north and south poles of two permanent magnets held fixed. Can we hope to generate current in the loop by using very strong magnets?

(b) A closed loop moves normal to the constant electric field between the plates of a large capacitor. Is a current induced in the loop
(i) when it is wholly inside the region between the capacitor plates
(ii) when it is partially outside the plates of the capacitor? The electric field is normal to the plane of the loop.
(c) A rectangular loop and a circular loop are moving out of a uniform magnetic field region (Fig. 6.8) to a field-free region with a constant velocity vv. In which loop do you expect the induced emf to be constant during the passage out of the field region? The field is normal to the loops.
(d) Predict the polarity of the capacitor in the situation described by Fig. 6.9.
Figure 6.8 — Illustration for Example 6.5 — a rectangular loop and a circular loop moving out of a uniform field region with constant velocity v.
Figure 6.8
Figure 6.9 — Illustration for Example 6.5 — predicting the polarity of the capacitor between two approaching magnets.
Figure 6.9
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Induction needs a changing magnetic flux, so: (a) No — a static field induces nothing however strong;

(b) No current in either case — an electrostatic field's changing electric flux cannot drive a loop current;

(c) the rectangular loop gives a constant emf;

(d) plate A is positive.

Guiding principle. Faraday's law, ε=− dΦB/dt\varepsilon=-\,d\Phi_B/dt with ΦB=∫B⃗⋅dA⃗\Phi_B=\int\vec{B}\cdot d\vec{A}, says an emf appears in a loop only when the magnetic flux linked with it changes. An electrostatic field is conservative, ∮E⃗⋅dl⃗=0\oint\vec{E}\cdot d\vec{l}=0, so it can never drive a steady current around a closed loop.

  1. Stationary loop, stationary magnets. Nothing moves, so B⃗\vec{B} and the loop's area are both fixed and ΦB\Phi_B is constant:

    ε=−dΦBdt=0.\varepsilon=-\frac{d\Phi_B}{dt}=0.

    Using stronger magnets only raises the (still constant) flux; it does not make it change. No current is generated.
  2. Loop moving in the constant field of a capacitor. Between the plates there is only an electric field (electrostatic, uniform); there is no magnetic field, so ΦB=0\Phi_B=0 at every instant and dΦB/dt=0d\Phi_B/dt=0.
    1. Wholly inside — the loop links no magnetic flux, so no current.
    2. Partly outside — now the electric flux through the loop does change, but a changing electrostatic flux does not induce a current in a conducting loop; only a changing magnetic flux does. So again there is no current. Hence no current is induced in either position.
  3. Rectangular vs circular loop leaving the field. With BB normal to the loops, ε=−B dA/dt\varepsilon=-B\,dA/dt, where AA is the area still inside the field.
  • Rectangular loop: the edge parallel to v⃗\vec{v} has a fixed length ℓ\ell, so A=ℓxA=\ell x and dA/dt=−ℓvdA/dt=-\ell v is constant — hence ∣ε∣=Bℓv|\varepsilon|=B\ell v is constant while it exits. …
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