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(C) Since the wires are connected in parallel across the same battery,the potential difference $V$ across each wire is the same. The current in each w

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$5 : 3$

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(C) Since the wires are connected in parallel across the same battery,the potential difference $V$ across each wire is the same. The current in each wire is given by $i = V/R$. Given the ratio of resistances $R_1 : R_2 : R_3 = 3 : 4 : 5$,the currents are in the ratio $i_1 : i_2 : i_3 = 1/3 : 1/4 : 1/5$.Let $i_1 = k/3$,$i_2 = k/4$,and $i_3 = k/5$,where $k$ is a constant.The force per unit length between two parallel wires carrying currents $i_a$ and $i_b$ separated by distance $r$ is $F = \frac{\mu_0 i_a i_b}{2\pi r}$.For the middle wire to experience zero net force,the force from the top wire must equal the force from the bottom wire: $\frac{\mu_0 i_1 i_2}{2\pi r_1} = \frac{\mu_0 i_3 i_2}{2\pi r_2}$.Simplifying this,we get $\frac{i_1}{r_1} = \frac{i_3}{r_2}$,which implies $\frac{r_1}{r_2} = \frac{i_1}{i_3}$.Substituting the values: $\frac{r_1}{r_2} = \frac{k/3}{k/5} = \frac{5}{3}$.Thus,the ratio of distances is $5 : 3$.

Three long straight wires are connected in parallel to each other across a battery of negligible internal resistance. The ratio of their resistances is $3 : 4 : 5$. What is the ratio of the distances of the middle wire from the others if the net force experienced by it is zero?

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