Съдържание

IPhO 1970, Теория и експеримент — Задача 3. Problem 3.

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IPhO 1970

Problems of the IV International Olympiad, Moscow, 1970

The publication is prepared by Prof. S. Kozel & Prof. V.Orlov (Moscow Institute of Physics and Technology)

The IV International Olympiad in Physics for schoolchildren took place in Moscow (USSR) in July 1970 on the basis of Moscow State University. Teams from 8 countries participated in the competition, namely Bulgaria, Hungary, Poland, Romania, Czechoslovakia, the DDR, the SFR Yugoslavia, the USSR. The problems for the theoretical competition have been prepared by the group from Moscow University stuff headed by professor V.Zubov. The problem for the experimental competition has been worked out by B. Zvorikin from the Academy of Pedagogical Sciences.

It is pity that marking schemes were not preserved.

Условие

Inside a thin-walled metal sphere with radius $R=20\ \mathrm{cm}$ there is a metal ball with the radius $r = 10\ \mathrm{cm}$ which has a common centre with the sphere. The ball is connected with a very long wire to the Earth via an opening in the sphere (Fig. 3). A charge $Q = 10^{-8}\ \mathrm{C}$ is placed onto the outside sphere. Calculate the potential of this sphere, electrical capacity of the obtained system of conducting bodies and draw out an equivalent electric scheme.

Two concentric circles: the inner ball of radius r and the outer sphere of radius R; a wire from the ball goes down through an opening in the sphere to an earth (ground) symbol. — Fig. 3

Отговори

Решение

Having no charge on the ball the sphere has the potential

\varphi_{0\mathrm{s}} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{R} = 450\mathrm{V}.

When connected with the Earth the ball inside the sphere has the potential equal to zero so there is an electric field between the ball and the sphere. This field moves a certain charge $q$ from the Earth to the ball. Charge $Q'$ , uniformly distributed on the sphere, doesn’t create any field inside thus the electric field inside the sphere is defined by the ball’s charge $q$ . The potential difference between the balls and the sphere is equal

\Delta\varphi = \varphi_{\mathrm{b}} - \varphi_{\mathrm{s}} = \frac{1}{4\pi\varepsilon_0}\left(\frac{q}{r} - \frac{q}{R}\right), \qquad (1)

Outside the sphere the field is the same as in the case when all the charges were placed in its center. When the ball was connected with the Earth the potential of the sphere $\varphi_{\mathrm{s}}$ is equal

\varphi_{\mathrm{s}} = \frac{1}{4\pi\varepsilon_0}\frac{q+Q}{R}. \qquad (2)

Then the potential of the ball

\varphi_{\mathrm{b}} = \varphi_{\mathrm{s}} + \Delta\varphi = \frac{1}{4\pi\varepsilon_0}\left(\frac{q+Q}{R} + \frac{q}{r} - \frac{q}{R}\right) = \frac{1}{4\pi\varepsilon_0}\left(\frac{Q}{R} + \frac{q}{r}\right) = 0 \qquad (3)

Which leads to

q = -Q\frac{r}{R}. \qquad (4)

Substituting (4) into (2) we obtain for potential of the sphere to be found:

\varphi_{\mathrm{s}} = \frac{1}{4\pi\varepsilon_0}\frac{Q - Q\frac{r}{R}}{R} = \frac{1}{4\pi\varepsilon_0}\frac{Q(R-r)}{R^2} = 225\mathrm{V}.

The electric capacity of whole system of conductors is

C = \frac{Q}{\varphi_{\mathrm{s}}} = \frac{4\pi\varepsilon_0 R^2}{R-r} = 4.4\cdot 10^{-11}\mathrm{F} = 44\mathrm{pF}

The equivalent electric scheme consists of two parallel capacitors: 1) a spherical one with charges $+q$ and $-q$ at the plates and 2) a capacitor “sphere – Earth” with charges $+(Q-q)$ and $-(Q-q)$ at the plates (Fig.5).

Equivalent circuit: two capacitors connected in parallel between a terminal on the left and an earthed node on the right; the upper (spherical) capacitor carries +q and −q, the lower one carries +(Q−q) and −(Q−q). — Fig. 5

Acknowledgement

The authors would like to thank Professor Waldemar Gorzkowski (Poland) and Professor Ivo Volf (Czech Republic) for their providing the materials of the IV IPhO in the Polish and Czech languages.

References:

O.Kabardin, V.Orlov, International Physics Olympiads for Pupils, Nauka, Moskva 1985
W.Gorzkowski, Zadania z fiziki z calego swiata – 20 lat. Miedzyna – rodowych Olimpiad Fizycznych, WNT, Warszawa 1994
V.Urumov, Prosventno Delo, Skopje 1999
D.Kluvanec, I.Volf, Mezinarodni Fysikalni Olympiady (metodycky material), MaFy, Hradec Kralowe 1993

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