Short on time? This page is the recap. For the derivations, Figure 1 and the worked examples, use the full study guide. This page is for the calculus-based course.
Recap
- When a charge q moves from potential V_i to V_f, the potential energy of the charge–field system changes by ΔU = q(V_f − V_i).
- If only the electric force does work, K + U is constant, so the kinetic energy changes by ΔK = −ΔU.
- With other forces, W_ext = ΔK + ΔU.
- The electrostatic force is conservative: the path does not matter, only the end potentials.
- A charge released from rest moves so that U decreases.
- 1 eV = 1.602 × 10⁻¹⁹ J, the energy one elementary charge gains through 1 V.
Key relationships
| Quantity | Relationship | Note |
|---|---|---|
| Potential energy change | ΔU = qΔV | keep the signs of q and ΔV |
| Kinetic energy change (electric force only) | ΔK = −qΔV | gain when qΔV < 0 |
| Work done by the field | W_field = −ΔU = −qΔV | |
| Speed from rest | v = √(2|q||ΔV|/m) | v ∝ √ΔV and v ∝ √(q/m) |
| Potential difference from a field | ΔV = −∫E·dr | then use ΔU = qΔV |
| Turning point | U = total energy, K = 0 | charge reverses there |
| Two free charges | Σp constant and K_total = −ΔU | lighter object gets more K |
Assumptions
- The field is static (charges that create it do not move, or move as part of the system you analyse).
- No energy is lost to friction, collisions or radiation; the motion is in a vacuum.
- Speeds are well below the speed of light, so K = ½mv².
Mistakes to avoid
- Losing the sign of q. Electrons gain kinetic energy moving to higher V.
- Using V, not ΔV. Only differences matter.
- Thinking speed ∝ ΔV. K ∝ ΔV; v ∝ √ΔV.
- Forgetting to convert eV to J before using ½mv².
- Using a plate separation you do not need. Energy depends only on ΔV.
- Giving all the energy to one body when both charges are free to move.
Quick self-check
- A particle of charge +2e is accelerated from rest through 500 V. What kinetic energy does it gain? (1000 eV, or 1.60 × 10⁻¹⁶ J)
- The accelerating potential difference is made 9 times larger. By what factor does the final speed change? (3 times, since v ∝ √ΔV)
- An electron moves from 0 V to +150 V, starting from rest. What is its kinetic energy? (150 eV = 2.40 × 10⁻¹⁷ J)
- A proton has 50 eV in total and the potential rises to +60 V ahead of it (U = 0 where V = 0). Does it get through? (No: it turns back where V = +50 V)
Next: practice questions.