C

Step 1: Identify the variables in the problem:
- Length of rods: \( \lambda \)
- Masses: \( m_1 \) and \( m_2 \)
- Force: \( F \)
- Magnetic field: \( B \)
Step 2: As the rods are pulled apart by a constant force \( F \), they will accelerate. Use Newton's second law: \( F = m a \) where \( a \) is the acceleration. The acceleration can be expressed as:
\( a = \frac{F}{m_{total}} = \frac{F}{m_1 + m_2} \)
Step 3: Velocity as a function of time can be derived from acceleration:
\( v(t) = a t = \frac{F}{m_1 + m_2} t \)
Step 4: The magnetic flux \( \Phi \) linked with the area swept by the rods can be expressed as: \( \Phi = B A = B(\lambda v) = B \lambda \left(\frac{F}{m_1 + m_2} t \right) \)
Step 5: The current induced in the circuit can be calculated using Faraday's law of induction, \( \mathcal{E} = -\frac{d\Phi}{dt} \). Taking the derivative:
\( \mathcal{E} = -B \lambda \frac{F}{m_1 + m_2} \)
Step 6: Using Ohm's law to find current: \( I = \frac{\mathcal{E}}{R} = \frac{-B \lambda \frac{F}{m_1 + m_2}}{R} \)
Therefore, the current in the resistor R as a function of time is independent of time and can be expressed as:
\( I = \frac{B \lambda F}{R (m_1 + m_2)} \).

Hence, the correct option is C.