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Physics LibreTexts

1.6.4: Field on the Axis of and in the Plane of a Charged Ring

Field on the axis of a charged ring.

Ring, radius a, charge Q. Field at P from element of charge \(δQ = \frac{\delta Q}{4\pi\epsilon_0 (a^2+z^2)}\). Vertical component of this \(= \frac{\delta Q \cos theta}{4\pi\epsilon_0 (a^2+z^2)}=\frac{\delta Qz}{4\pi\epsilon_0 (a^2+z^2)^{3/2}}\). Integrate for entire ring:

Field \(E = \frac{Q}{4\pi\epsilon_0}\frac{z}{(a^2+z^2)^{3/2}}\).

In terms of dimensionless variables:

\(E=\frac{z}{(1+z^2)^{3/2}}\),

where E is in units of \(\frac{Q}{4\pi\epsilon_0 a^2}\), and z is in units of a.

From calculus, we find that this reaches a maximum value of \(\frac{2\sqrt{3}}{9}=0.3849\) at \(z=1/\sqrt{2}=0.7071\).

It reaches half of its maximum value where \(\frac{z}{(1+z^2)^{3/2}}=\frac{\sqrt{3}}{9}\).

That is, \(3-72Z+9Z^2+3Z^2=0\), where \(Z=z^2\).

The two positive solution are \(Z = 0.041889 \text{ and }3.596267\).

That is, \(z = 0.2047 \text{ and }1.8964\).

 


Field in the plane of a charged ring.

We suppose that we have a ring of radius a bearing a charge Q. We shall try to find the field at a point in the plane of the ring and at a distance \(r (0 ≤ r < a)\) from the centre of the ring.

Consider an element δθ of the ring at P. The charge on it is \(\frac{Q\delta \theta}{2\pi}\). The field at A due this element of charge is

\(\frac{1}{4\pi\epsilon_0}\cdot \frac{Q\delta\theta}{2\pi}\cdot \frac{1}{a^2+r^2-2ar\cos \theta}=\frac{Q}{4\pi\epsilon_0 .2\pi a^2}\cdot \frac{\delta\theta}{b-c\cos \theta}\),

where \(b=1+r^2/a^2\) and \(c = 2r / a\). The component of this toward the centre is

\(-\frac{Q}{4\pi\epsilon_0 .2 \pi a^2}\cdot\frac{\cos \phi \delta\theta}{b-c\cos \theta}\).

To find the field at A due to the entire ring, we must express \(\phi\) in terms of θ, r and a, and integrate with respect to \(θ \text{ from }0 \text{ to }2π\) (or from \(0 \text{ to }π\) and double it). The necessary relations are

\[p^2=a^2+r^2-2ar\cos \theta\]

\[\cos \phi = \frac{r^2+p^2-a^2}{2rp}.\]

The result of the numerical integration is shown below, in which the field is expressed in units of \(Q/(4\pi\epsilon_0 a^2)\) and r is in units of a.