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