Irodov 3.61: A Thread Standing on Its Image – Amit & Deepali Sinha Foundation

Problem Irodov 3.61

A very long straight thread stands at right angles to an infinite conducting plane, with its near end a distance $l$ from the plane. The thread carries a uniform charge $\lambda$ per unit length. Let $O$ be the point where the line of the thread meets the plane. Find the surface density of the charge induced on the plane

at the point $O$ ;
as a function of the distance $r$ from $O$ .

The thread with charge λ per unit length, normal to the conducting plane, its near end at distance l (it continues upward without end); O is its trace on the plane, and a point of the plane lies at distance r from O. — The thread with charge $\lambda$ per unit length, normal to the conducting plane, its near end at distance $l$ (it continues upward without end); $O$ is its trace on the plane, and a point of the plane lies at distance $r$ from $O$ .

Given

charge per length $\lambda$
thread normal to the plane
end at $l$ from it

Find

(a)

\sigma

O

· (b)

\sigma(r)

Solution

Step 1The image doubles the normal field

The plane is an equipotential, so above it the field is that of the thread and its mirror image: a thread $-\lambda$ on the same line, from depth $l$ downward without end. At a point $P$ of the plane each element of the image is the mirror of an element of the thread, of opposite sign and equally far away. The two fields at $P$ therefore have equal components along the plane’s normal $z$ and opposite components along the plane, so only twice the thread’s normal component survives.

E_z = 2E_z^{\text{thread}}, \qquad E_r = 0 \tag{1}

Step 2The thread’s normal field at

P

Take $P$ at distance $r$ from $O$ and an element $dq = \lambda\,dz$ of the thread at height $z$ , a distance $s = \sqrt{r^2 + z^2}$ from $P$ .

The field of the thread alone at a point P of the plane's trace (dashed): an element dq at height z, its distance s to P, its field d E and the components dE_r along the plane and dE_z along the normal. — The field of the thread alone at a point $P$ of the plane’s trace (dashed): an element $dq$ at height $z$ , its distance $s$ to $P$ , its field $d\vect E$ and the components $dE_r$ along the plane and $dE_z$ along the normal.

For $\lambda > 0$ the element’s field points away from it, and its component along the normal points into the plane, a fraction $z/s$ of the whole.

dE_z^{\text{thread}} = -\frac{1}{4\pi\varepsilon_0}\frac{\lambda\,dz}{s^2}\cdot\frac{z}{s} = -\frac{\lambda}{4\pi\varepsilon_0}\frac{z\,dz}{(r^2 + z^2)^{3/2}}

The thread runs from $z = l$ to infinity.

E_z^{\text{thread}} = -\frac{\lambda}{4\pi\varepsilon_0}\int_l^\infty \frac{z\,dz}{(r^2 + z^2)^{3/2}} = -\frac{\lambda}{4\pi\varepsilon_0}\left[-\frac{1}{\sqrt{r^2 + z^2}}\right]_l^\infty

E_z^{\text{thread}} = -\frac{\lambda}{4\pi\varepsilon_0\sqrt{l^2 + r^2}}

(2)

Step 3(b) The induced density at distance

r

At a conductor’s surface $\sigma = \varepsilon_0 E_n$ , with $E_n$ the component along the outward normal, here $+z$ . By (1) and (2):

\sigma = \varepsilon_0\cdot 2E_z^{\text{thread}} = -\frac{2\varepsilon_0\lambda}{4\pi\varepsilon_0\sqrt{l^2 + r^2}}

\sigma(r) = -\dfrac{\lambda}{2\pi\sqrt{l^2 + r^2}}

(3)

Step 4(a) The density at

O

The point $O$ is $r = 0$ in (3).

\sigma(O) = -\dfrac{\lambda}{2\pi l}

(4)

The induced charge has the sign opposite to $\lambda$ ; its magnitude is largest at $O$ and falls off only as $1/r$ .

Check

Units: $\lambda/l$ is a charge per area. A limit with a known answer: with $l \to 0$ the point $P$ lies on the perpendicular through the thread’s end, at distance $r$ , where a semi-infinite thread’s field has its component along the thread equal to $\lambda/(4\pi\varepsilon_0 r)$ ; (2) gives exactly that.

\lim_{l \to 0}\left|E_z^{\text{thread}}\right| = \frac{\lambda}{4\pi\varepsilon_0 r}

Remark

The induced charge inside a circle of radius $R$ about $O$ has a strikingly simple form.

q_{\text{in}}(R) = \int_0^R \sigma\,2\pi r\,dr = -\lambda\int_0^R \frac{r\,dr}{\sqrt{l^2 + r^2}} = -\lambda\left(\sqrt{l^2 + R^2} - l\right)

It is minus the charge on the stretch of thread from its end out to the height $\sqrt{l^2 + R^2}$ , and it grows without bound as $R$ does: an infinite thread induces an infinite charge, spread so thinly that the density falls only as $1/r$ .