Two conservation laws together

Problems involving two charges interacting without external forces — released from rest, or fired at each other — have a recurring structure: they bring two conservation laws into play at once. Recognising both is the key to solving them.

The two conservations

• Momentum (a vector quantity): conserved because no external forces act on the system of the two charges. The Coulomb force is internal and, by Newton’s third law, does not alter the total momentum.
• Energy (a scalar quantity): conserved because the Coulomb force is conservative and admits the potential energy $E_{\text{pot,el}} = k\,q_1 q_2 / r$.

This is exactly the same scheme as the method of two parallel tables already used for mechanical collisions: one table for momentum (before/after), one for energy (before/after). Conservation of momentum links the velocities of the two particles together — for example, for two masses initially at rest, it imposes $m_1\vv{v}_1 + m_2\vv{v}_2 = 0$, i.e. $\vv{v}_2 = -\dfrac{m_1}{m_2}\vv{v}_1$ — while conservation of energy fixes the magnitude of the velocities from the change in $E_{\text{pot,el}}$ between the two configurations.

The two equations together are enough to fully determine the final state. The value of this method lies precisely in how they interlock: neither momentum conservation alone (which gives the ratios between the velocities but not their magnitudes), nor energy conservation alone (which gives the total energy but not how it is shared out), would solve the problem by itself. It is their combination that closes it.

Links

Topics: Electrostatics Concepts: Electrostatic potential energy · Conservation of momentum Skills: Method of the two tables Methods: Method of the two parallel tables

Related exercises: Problem — Two charges released from rest · Problem — The two gunslingers · Problem — Two charges launched upward