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Consider real numbers X1,X2 ......Xn, and Q1,Q2 ......Qn. ALL X's are not zero. Similarly, ALL Q's are not zero.

If

(i) X1+X2+........+Xn=0

    and if

(ii) For every X belonging to the set {X1,X2.....,Xn}, -X necessarily belongs to the same set

   and if

(iii) X1Q1 + X2Q2 + .......... + XnQn=0

    and if

(iv) X1(Q1)^2 + X2(Q2)^2 + ........ + Xn(Qn)^2=0,

Can we conclude that Q1=Q2=Q3=........=Qn?

(I am trying to prove that the surface charge density on a conducting, isolated sphere is uniform)

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  • $\begingroup$ If you allow negative $x_i$, then it is easy to get solutions with unequal $q_i$ for $n>3$. If you fix the $q_i$ in an arbitrary way, then you only have three equations for $n>3$ variables $x_i$ and in general there will be infinitely many solutions. $\endgroup$ – almagest Jun 16 '16 at 13:03
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In the case $\forall i \in \{1,\cdots,n\}\;\; X_i = 0$, any set of of values of $Q_i$ wil satisfy your three conditions. Which gives a counter example.

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