Deriving I = U/R from the microscopic drift model

Step 1 of 4: The field accelerates electrons

Intuition

The electric field is like an invisible slope: electrons 'roll downhill' in potential — the steeper the slope (larger U, shorter wire), the faster they roll.

E = \frac{U}{L}, \qquad a = \frac{eE}{m_e}

Analysis

A voltage $U$ across a conductor of length $L$ sets up a uniform field $E = U/L$ . Each free electron (charge $e$ , mass $m_e$ ) feels the force $eE$ and accelerates at $a = eE/m_e$ — Newton's second law applied to the electron.