Mathematical and Physical Journal
for High Schools
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Problem P. 5095. (January 2019)

P. 5095. Two resistors of resistance values \(\displaystyle R_{1}\) and \(\displaystyle R_{2}\) are connected in series and their equivalent resistance is \(\displaystyle R_{1}+R_{2}\). Two other resistors of resistance \(\displaystyle R\) are connected in the circuit, one of them and \(\displaystyle R_{1}\) are connected in parallel, whilst the other one and \(\displaystyle R_{2}\) are connected in series. Is there a value of \(\displaystyle R\) such that the equivalent resistance of the whole circuit remains \(\displaystyle R_{1}+R_{2}\)?

(4 pont)

Deadline expired on February 11, 2019.


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

Megoldás. A megadott feltétel szerint

\(\displaystyle \left(\frac{1}{R_1}+\frac{1}{R}\right)^{-1}+R_2+R=R_1+R_2,\)

vagyis

\(\displaystyle \frac{R_1R}{R_1+R}+R=R_1.\)

Bevezetve az \(\displaystyle x=R/R_1\) jelölést, a fenti összefüggés az

\(\displaystyle x^2+x-1=0\)

másodfokú egyenletre vezet, aminek pozitív gyöke:

\(\displaystyle x=\frac{\sqrt{5}-1}{2}\approx 0{,}62.\)

A keresett ellenállás nagysága tehát \(\displaystyle R_2\)-től függetlenül \(\displaystyle R\approx 0{,}62\,R_1.\)


Statistics:

91 students sent a solution.
4 points:76 students.
3 points:8 students.
2 points:4 students.
1 point:1 student.
0 point:2 students.

Problems in Physics of KöMaL, January 2019