Solution
Circuit Analysis:
The battery is connected between two points on the ring. The ring acts as two resistors in parallel. Let the points divide the ring into two arcs of resistance $R_1$ and $R_2$. For maximum symmetry and force, we consider diametrically opposite points.
- $R_1 = R_2 = R/2$
- Equivalent Resistance $R_{eq} = (R/2)/2 = R/4$
- Total Current $I = \mathcal{E} / R_{eq} = 4\mathcal{E}/R$
Magnetic Force Calculation:
The total current $I$ enters at one point and leaves at the other. The magnetic force on a current carrying loop segment depends on the effective length (chord length) connecting the terminals.
For diametrically opposite points, effective length $L = 2r$.
$$ F_{max} = I L B = \left( \frac{4\mathcal{E}}{R} \right) (2r) B $$
$$ F_{max} = \frac{8rB\mathcal{E}}{R} $$
Configuration: Connect the battery between two points at the ends of a diameter perpendicular to the magnetic field.