Mathematical and Physical Journal
for High Schools
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# Problem P. 4192. (October 2009)

P. 4192. The metal (hollow) spheres of radii R, 2R and 3R are placed into each other such that their centres are at the same point. The inner sphere is given a charge of Q, the middle one is charged to 2Q, and the outer one is charged to 3Q. a) Find the potentials, measured from the common centre of the circles, at a distances of R, 2R, 3R and 4R, if the potential at the centre is taken to be zero. b) What is the energy of the electric field in the regions between the spheres and outside the spheres? (R=10 cm, Q=2.10-6 C.)

(5 pont)

Deadline expired on November 10, 2009.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. $\displaystyle a)$ A potenciálok:

$\displaystyle U(R)=0, \quad U(2R)=-k\frac{Q}{2R}=-90~{\rm kV},\\ \quad U(3R)=-k\frac{Q}{R}=-180~{\rm kV}, \quad U(4R)=-k\frac{3Q}{2R}=-270~{\rm kV}.$

$\displaystyle b)$ Az elektrosztatikus térenergiák:

$\displaystyle W(0<r<R)=0,\quad W(R<r<2R)=\frac{1}{4}\, \frac{kQ^2}{R}=0{,}09~{\rm J}, \quad W(2R<r<3R)=\frac{3}{4}\, \frac{kQ^2}{R}=0{,}27~{\rm J}, \quad W(3R<r)=6\, \frac{kQ^2}{R}=2{,}16~{\rm J}.$

### Statistics:

 35 students sent a solution. 5 points: Galzó Ákos Ferenc. 4 points: Laczkó Zoltán Balázs. 3 points: 11 students. 2 points: 8 students. 1 point: 8 students. 0 point: 5 students. Unfair, not evaluated: 1 solution.

Problems in Physics of KöMaL, October 2009