A current carrying rectangular loop $$PQRS$$ is made of a uniform wire. The length $$PR = QS = 5$$ cm and $$PQ = RS = 100$$ cm. If the ammeter current reading changes from $$I$$ to $$2I$$, then the ratio of the magnetic forces per unit length on the wire $$PQ$$ due to the wire $$RS$$ in the two cases respectively $$(f^I_{PQ} : f^{2I}_{PQ})$$ is:

$$i_1 = \frac{I}{2}$$ and $$i_2 = \frac{I}{2}$$

Force per unit length $$f_1 = \frac{\mu_0 (\frac{I}{2}) (\frac{I}{2})}{2\pi d} = \frac{\mu_0 I^2}{8\pi d}$$

$$i_1 = \frac{2I}{2} = I$$ and $$i_2 = \frac{2I}{2} = I$$

Force per unit length $$f_2 = \frac{\mu_0 (I) (I)}{2\pi d} = \frac{\mu_0 I^2}{2\pi d}$$

$$\frac{f_1}{f_2} = \frac{\frac{\mu_0 I^2}{8\pi d}}{\frac{\mu_0 I^2}{2\pi d}} = \frac{2}{8} = \frac{1}{4}$$