= Solution
Compare the currents using the \b[same calibrated detector and unchanged geometry]. With a <tangent galvanometer>, a circular coil gives $B_c=\mu_0NI/(2a)$ at its centre, perpendicular to the horizontal terrestrial field $B_H$. A small compass needle aligns with their resultant, so
$$
\tan\theta=\frac{B_c}{B_H},\qquad
\boxed{\frac{I_1}{I_2}=\frac{\tan\theta_1}{\tan\theta_2}.}
$$
Use the same coil and location, set its plane in the magnetic meridian, and reverse each current to help identify zero offsets or unwanted local fields. The needle must be sufficiently small relative to the coil for the nearly uniform central field approximation. A finite deflection is not generally proportional to current; replacing the tangent ratio by the angle ratio requires a small-angle approximation or a different calibrated instrument.
<Electrolysis> offers another comparison. With identical suitable <voltameters>, the same reaction and constant currents acting for equal times, the deposited mass or collected reaction product is proportional to charge $Q=It$, according to <Faraday's laws of electrolysis>. Thus $I_1/I_2=m_1/m_2$ when the same product is collected with equal current efficiency. For unequal observation times, compare $m_1/t_1$ with $m_2/t_2$. Gas collection additionally requires matching pressure, temperature and vapour corrections. Current-dependent side reactions can invalidate a naive product ratio.
An essential circuit qualification is that adding the detector can change the current. Either compare the currents in the circuits as actually measured or account for detector resistance and source loading. Comparing two voltaic sources' currents through one specified circuit is not the same as establishing a ratio of their <electromotive forces>.
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