B

To find an expression ensuring the galvanometer shows zero deflection, we need to set the currents through the two inductors such that the net magnetic effect at the galvanometer is zero at all times after the key is closed.
Step 1: When the key K is closed at t = 0, inductors L1 and L2 will oppose the change in current due to their self-inductance. This induces an electromotive force (emf) that will affect the circuit.
Step 2: Applying Kirchhoff's loop rule to the circuit, we have:
$$ V - I_1 R_1 - L_1 \frac{dI_1}{dt} = 0 $$
$$ V - I_2 R_2 - L_2 \frac{dI_2}{dt} = 0 $$
where V is the applied voltage, I1 is the current through L1, and I2 is the current through L2.
Step 3: To ensure the galvanometer shows zero deflection, the currents I1 and I2 must be equal, i.e., I1 = I2 at all times.
Step 4: Thus, we can equate the expressions from the above currents:
$$ I_1 = \frac{V}{R_1} e^{-\frac{R_1}{L_1}t} $$
$$ I_2 = \frac{V}{R_2} e^{-\frac{R_2}{L_2}t} $$
Therefore, for zero deflection,
$$ \frac{V}{R_1} = \frac{V}{R_2} $$
which implies R1 = R2.
Conclusion: Therefore, the condition for the galvanometer to show zero deflection at all times is that the resistances R1 and R2 should be equal.