Diagonal voltages

Electricity and Magnetism Level 2

In the circuit below, E = 42 V E=\SI{42}{\volt} , the potential difference between points A A and C C is U A C = 39 V U_{AC}=\SI{39}{\volt} , and that between points B B and D D is U B D = 21 V U_{BD}=\SI{21}{\volt} . If resistor R 2 R_2 has a resistance of 60 Ω \SI{60}{\ohm} , find the values in ohms of R 1 R_1 and R 3 R_3 and enter your answer as their product.


The answer is 700.

Novak Radivojević by Novak Radivojević , 23, Serbia

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1 solution

Chew-Seong Cheong
Aug 18, 2017

From the circuit, we have:

{ U A D = 42 V U A C = 39 V U B D = 21 V ⟹ { U A B = U A D − U B D = 42 − 21 = 21 V U C D = U A D − U A C = 42 − 39 = 3 V U B C = U B D − U C D = 21 − 3 = 18 V \begin{cases} U_{AD} = 42 \text{ V} \\ U_{AC} = 39 \text{ V} \\ U_{BD} = 21 \text{ V} \end{cases} \implies \begin{cases} U_{AB} = U_{AD} - U_{BD} = 42 - 21 = 21 \text{ V} \\ U_{CD} = U_{AD} - U_{AC} = 42 - 39 = 3 \text{ V} \\ U_{BC} = U_{BD} - U_{CD} = 21 - 3 = 18 \text{ V} \end{cases}

Since resistors in series is a voltage divider, we have:

R 1 : R 2 : R 3 = U A B : U B C : U C D ⟹ R 1 : 60 : R 3 = 21 : 18 : 3 ⟹ R 1 = 21 18 × 60 = 70 Ω ⟹ R 3 = 3 18 × 60 = 10 Ω \begin{aligned} R_1 : R_2 : R_3 & = U_{AB} : U_{BC} : U_{CD} \\ \implies R_1 : 60 : R_3 & = 21 : 18 : 3 \\ \implies R_1 & = \frac {21}{18} \times 60 = 70 \ \Omega \\ \implies R_3 & = \frac {3}{18} \times 60 = 10 \ \Omega \end{aligned}

⟹ R 1 R 3 = 700 \implies R_1R_3 = \boxed{700}

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