Short on time? This page is the recap. For Figure 1, the derivations and the worked examples, use the full study guide. This guide is for the calculus-based course.
Recap
- A moving charge makes a magnetic field. It depends on the velocity and on the distance to the point. B is perpendicular to both v and r, greatest when v ⊥ r and zero along the line of motion. Field lines circle the line of motion.
- A magnetic field pushes on a moving charge with a force perpendicular to both v and B. A charge at rest, or one moving along B, feels no magnetic force.
- For a negative charge, reverse the direction you get from the right-hand rule.
- The magnetic force does no work (F·v = 0), so speed is constant. In a uniform field the path is a circle (v ⊥ B) or a helix (v at an angle to B).
- With E and B together, the forces are independent and add as vectors.
- The Hall effect: in a field, carriers are pushed to one edge of a conductor, creating a potential difference across its width.
Key relationships
| Quantity | Relationship |
|---|---|
| Field of a moving point charge | B = (μ₀/4π) q (v × r̂)/r²; size (μ₀/4π)qv sin θ/r² |
| Magnetic force | F = q(v × B); size qvB sin θ |
| Combined fields | F = qE + q(v × B) |
| Radius and period in uniform B | r = mv/(qB); T = 2πm/(qB) |
| Helix pitch | v∥ × T |
| Undeflected speed in crossed E and B | v = E/B |
| Hall potential difference | ΔV_H = v_dBw = IB/(nqt) |
In the sizes and in r and T, q means the magnitude of the charge. μ₀/4π = 1 × 10⁻⁷ T·m/A; 1 T = 1 N/(A·m).
Assumptions
- Speeds much less than the speed of light (for the point-charge field and r = mv/(|q|B)).
- Uniform fields unless stated; gravity is negligible for charged particles.
- In the Hall derivation, all carriers drift at the same speed and the field is perpendicular to the strip.
Mistakes to avoid
- Wrong order in a cross product. v × B, not B × v; and v × r̂ with r from the charge to the point.
- Ignoring the sign of q. Electrons: reverse.
- Saying B speeds the charge up. It only turns it.
- Putting v in the period. T = 2πm/(|q|B).
- Using the whole speed for the radius of a helix. Only v⊥ goes into r.
- Mixing up width and thickness in the Hall formula. ΔV_H = IB/(nqt) uses the thickness t along B.
Quick self-check
- A proton moves at 2.0 × 10⁶ m/s at 30° to a 0.50 T field. What is the force on it? (8.0 × 10⁻¹⁴ N)
- A particle’s speed is doubled and the field is doubled. What happens to the radius of its circle? (No change)
- Where is the field of a moving charge zero? (Everywhere on its line of motion, ahead and behind)
Next: practice questions.