Question

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?

Short answer

The three points on the curve corresponding to the four points of Fig.32-19 are 1a, 2b, 3c, and d.

Step-by-step solution

Given

Figure 32-19a and 32-19b.

‘a’ and ‘b’ are equidistant from the central axis. Also ‘c’ and‘d’ are equidistant from it.

Determining the concept

From the magnetic field formulas inside and outside the circular capacitor, determine the relationship between magnetic field and distance from the central axis. Find the three points on the curve that correspond to the four points of Fig. 32-19a by using this information and examining the graph and the provided figure.

The formula is as follows:

1. Inside a circular capacitor, the magnetic field is
   $\mathit{B}\mathbf{=}\mathbf{\left(}\frac{{\mathbf{\mu }}_{\mathbf{0}}{\mathbf{l}}_{\mathbf{d}}}{\mathbf{2}\mathbf{\pi }{\mathbf{R}}^{\mathbf{2}}}\mathbf{\right)}\mathit{r}$
   
2. Outside a circular capacitor, the magnetic field is
   $\mathit{B}\mathbf{=}\mathbf{\left(}\frac{{\mathbf{\mu }}_{\mathbf{0}}{\mathbf{l}}_{\mathbf{d}}}{\mathbf{2}\mathbf{\pi r}}\mathbf{\right)}$
   

Determining thethree points on the curve corresponding to the four points of Fig.32-19a

The magnetic field at a point inside the capacitor is given by

$B=\left(\frac{{\mathrm{\mu }}_0{\mathrm{l}}_{\mathrm{d}}}{2{\mathrm{\pi R}}^2}\right)r$

It implies that$B\propto r$.

The curve on which point 2 is present satisfies this condition, and there is only one point inside the capacitor ‘b.’

Hence, point 2 corresponds to point b.

The magnetic field at a point outside the capacitor is given by

$B=\left(\frac{{\mathrm{\mu }}_0{\mathrm{l}}_{\mathrm{d}}}{2\mathrm{\pi r}}\right)$

It implies that $B\propto \frac{1}{r}$.

The curves on which points 1 and 3 are present satisfy this relation.

Point '1' corresponds to point 'a' since points 'a' and 'b' are equally spaced apart.

Points 'c' and 'd' correspond to the last point 3, which is left.

Hence, the three points on the curve corresponding to the four points of Fig.32-19 are 1a, 2b, 3c, and d.

While outside of a circular capacitor, the magnetic field is inversely proportional to the distance from the circular plate's centre, inside a circular capacitor, it is proportional to that distance.