Chapter 4 / 4.3 Motion in a Magnetic Field

Motion in a Magnetic Field 🚀

1. Why the Path Bends but the Speed Stays the Same 🔄

When a charge zips through a magnetic field, the magnetic force always acts sideways to its motion, so it never does work. Speed stays constant while only direction changes — unlike an electric field, which can speed particles up or slow them down. :contentReference[oaicite:0]{index=0}

2. Pure Circular Motion ( v ⟂ B ) 🎯

Centripetal balance: the sideways magnetic force supplies the needed inward force: \( m\,\dfrac{v^{2}}{r}=q\,v\,B \) →
Radius: \( r=\dfrac{m\,v}{q\,B} \) 🌟 :contentReference[oaicite:1]{index=1}
Cyclotron (angular) frequency: \( \omega = \dfrac{q\,B}{m} \) and \( n=\dfrac{\omega}{2\pi}=\dfrac{q\,B}{2\pi m} \) — notice it does not depend on speed or energy, a key idea behind the cyclotron! ⚙️ :contentReference[oaicite:2]{index=2}
Time for one circle: \( T=\dfrac{2\pi}{\omega}=\dfrac{2\pi m}{qB} \) ⏱️

3. Helical Motion ( v has a component along B ) 🌀

If the particle also has a speed \( v_{||} \) along the field, that part marches straight ahead while the perpendicular part keeps the circular dance. The result: a beautiful helix! Pitch (distance moved along the field during one turn): \( p = v_{||}\,T = \dfrac{2\pi m\,v_{||}}{q\,B} \) :contentReference[oaicite:3]{index=3}

4. Worked Example 💡

Electron in a 6 × 10⁻⁴ T field (speed 3 × 10⁷ m s⁻¹, mass 9 × 10⁻³¹ kg, charge 1.6 × 10⁻¹⁹ C):

Radius: 0.28 m (28 cm)
Rotation frequency: 17 MHz
Kinetic energy: 2.5 keV

Calculation uses the formulas above — plug in the numbers and you’re done! 🎉 :contentReference[oaicite:4]{index=4}

5. NEET Flashcards 🔥

Radius in a uniform B: \( r=\dfrac{m\,v}{q\,B} \)
Cyclotron frequency (speed-independent!): \( n=\dfrac{q\,B}{2\pi m} \)
Helical pitch: \( p=\dfrac{2\pi m\,v_{||}}{q\,B} \)
Magnetic force does no work ⇒ speed stays constant
Bigger momentum ⇒ bigger circular path (useful for mass spectrometry)

Keep practicing — these ideas pop up in NEET year after year. You’ve got this! ✨