The figure 32-20 shows a circular region of radius in which a displacement current is directed out of the page. The magnitude of the density of this displacement current is , where is the radial distance .(a) What is the magnitude of the magnetic field due to displacement current at ?(b) What is the magnitude of the magnetic field due to displacement current at ?
When a conductor is placed in a region of changing magnetic field, it induces a displacement current that starts flowing through it as it causes the case of an electric field produced in the conductor region. According to Lenz law, the current flows through the conductor such that it opposes the change in magnetic flux through the area enclosed by the loop or the conductor. The magnitude of the magnetic field is due to the displacement current using the displacement current density which is non-uniformly distributed.
The magnetic field at a point inside the capacitor, (i)
where, is the magnetic field, is the magnetic permittivity constant, is the radial distance, is the displacement current, is the radius of the circular region.
The magnetic field at a point outside the capacitor, (ii)
Where, is the magnetic permittivity constant , is the radial distance, is the displacement current.
The current flowing in a given region for non-uniform electric field, (iii)
Where, is the current density of the material, is the radial distance of the circular region, is the differential form of the radial distance.
The displacement current density is non-uniform. Hence, the displacement current is determined by taking the integration over the closed path of radius and that is given using the given data in equation (i) as follows:
The integral is limited to . Hence, by taking in equation (i), the magnetic field can be determined as follows:
Therefore, the magnitude of the magnetic field due to displacement current at a radial distance is .
Figure 32-23 shows a face-on view of one of the two square plates of a parallel-plate capacitor, as well as four loops that are located between the plates. The capacitor is being discharged. (a) Neglecting fringing of the magnetic field, rank the loops according to the magnitude of along them, greatest first. (b) Along which loop, if any, is the angle between the directions of constant (so that their dot product can easily be evaluated)? (c) Along which loop, if any, is B constant (so that B can be brought in front of the integral sign in Eq. 32-3)?
Figure 32-24 shows three loop models of an electron orbiting counterclockwise within a magnetic field. The fields are non-uniform for models 1 and 2 and uniform for model 3. For each model, are (a) the magnetic dipole moment of the loop and (b) the magnetic force on the loop directed up, directed down, or zero?
Figure 32-19a shows a capacitor, with circular plates, that is being charged. Point a (near one of the connecting wires) and point b (inside the capacitor gap) are equidistant from the central axis, as are point c (not so near the wire) and point d (between the plates but outside the gap). In Fig. 32-19b, one curve gives the variation with distance r of the magnitude of the magnetic field inside and outside the wire. The other curve gives the variation with distance r of the magnitude of the magnetic field inside and outside the gap. The two curves partially overlap. Which of the three points on the curves correspond to which of the four points of Fig. 32-19a?
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