Problem — Continuity of B

Problem

For the capacitor of the previous problem ($R$, constant charging current $i$), derive from Ampère-Maxwell the field $\abs{\vv B}(r)$ for $r<R$ and for $r>R$. (a) Write the two expressions. (b) Show that they match up continuously at $r=R$. (c) Sketch $\abs{\vv B}(r)$ and compare it with a solid wire carrying a current.

Solution

(a) The two expressions. Inside ($r<R$) the circle encloses only the fraction $(r/R)^2$ of the total displacement current $i$; outside ($r>R$) it encloses the whole current $i$. Applying $\oint\vv B\cdot d\vv\ell=B\,(2\pi r)$:

$\ev{B=\frac{\mu_0\,i\,r}{2\pi R^2}}\qquad (r<R)$

$\ev{B=\frac{\mu_0\,i}{2\pi r}}\qquad (r>R)$

(b) Matching at $r=R$. Evaluating both at $r=R$:

$\frac{\mu_0\,i\,R}{2\pi R^2}=\frac{\mu_0\,i}{2\pi R}=\frac{\mu_0\,i}{2\pi R}$

The two expressions give the same value: the field is continuous, with no jumps.

(c) Graph and comparison. The field grows linearly from $0$ up to the peak $\dfrac{\mu_0 i}{2\pi R}$ at $r=R$, then decreases as $1/r$.

This is the same trend as the field of a solid straight wire carrying a uniform current: inside the conductor $B\propto r$, outside $B\propto 1/r$. The displacement current in the capacitor generates a magnetic field entirely analogous to that of a real current.

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Topics: Onde elettromagnetiche Concepts: Corrente di spostamento · Campo magnetico