Source: High school physics (Chinese)
Problem
Two resistance wires of the same material are connected in two separate circuits. Wire A has length $l$ and diameter $d$; wire B has length $3l$ and diameter $3d$.
To make both wires produce equal heat in the same time, what should be the ratio of voltages applied to them, $U_A : U_B$?
A. $U_A : U_B = 1 : 1$
B. $U_A : U_B = \sqrt{3} : 3$
C. $U_A : U_B = \sqrt{3} : 1$
D. $U_A : U_B = 3 : 1$
Resistance of a wire: $R = \dfrac{\rho L}{A} = \dfrac{4\rho L}{\pi d^2}$.
$$R_A = \frac{4\rho l}{\pi d^2}, \qquad R_B = \frac{4\rho (3l)}{\pi (3d)^2} = \frac{4\rho l}{3\pi d^2}$$So $\dfrac{R_A}{R_B} = 3$.
For equal heat in equal time, $\dfrac{U_A^2}{R_A} t = \dfrac{U_B^2}{R_B} t$, giving $\dfrac{U_A^2}{U_B^2} = \dfrac{R_A}{R_B} = 3$.
Therefore $U_A : U_B = \sqrt{3} : 1$. Answer: C.