Magnetic Fields and Forces on Moving Charges

Key Concepts

• Magnetic Field and Force:

  • A charge in a magnetic field experiences no force if it's not moving.
  • If a charge moves parallel to the magnetic field, there is also no force on the charge.
  • A charge experiences force only when it moves perpendicular to the magnetic field.

• Force Equation:

  • The force ( F ) on a moving charge ( Q ) is given by the equation:
    [ F = Q \cdot (V \times B) ]
  • Magnitude of the force:
    [ |F| = Q \cdot V \cdot B \cdot \sin(\theta) ]
  • ( \theta ) is the angle between the velocity ( V ) of the charge and the magnetic field ( B ).
  • If ( \theta = 90^\circ ), ( \sin(90^\circ) = 1 ), hence ( |F| = QVB ).
  • If ( \theta = 0^\circ ), ( \sin(0^\circ) = 0 ), resulting in no force.

Direction of Force

• Positive Charges: Use the right hand rule.
• Negative Charges: Use the left hand rule.
• Procedure:
  • Point fingers in the direction of velocity.
  • Curl fingers in the direction of the magnetic field.
  • Thumb points in the direction of the force.

Examples

1. Example 1:

   • Positive charge moving upwards through a horizontal magnetic field to the right.
   • Right hand rule: Force is into the board.

2. Example 2:

   • Negative charge moving right through the same magnetic field.
   • Left hand rule: Force is also into the board.

3. Example 3:

   • Positive charge moving right, magnetic field into the board.
   • Right hand rule: Force upwards.

4. Example 4:

   • Negative charge moving downward with magnetic field out of the board.
   • Left hand rule: Force to the right.

5. Example 5 & 6

   • When a charge moves in the same or directly opposite direction of the magnetic field, the force is zero (( \theta = 0^\circ ) or ( 180^\circ )).

Common Mistakes

• Forgetting to use the correct hand based on charge polarity.
• Incorrectly aligning the direction of the magnetic field.
• Misapplying the angle ( \theta ) in calculating force magnitude.

Conclusion

• Use the left/right hand rule to determine the direction of the force on moving charges in a magnetic field.
• Ensure the charge's movement is not parallel to the magnetic field for a force to exist.
• Upcoming: Calculation of actual magnitudes and directional forces in examples.