A moving electric charge can feel a magnetic force when it travels through a magnetic field. This force is central to devices such as particle accelerators, mass spectrometers, electric motors, and old-style television tubes. Unlike electric force, magnetic force depends on the charge velocity and its direction relative to the field.
The result is often a sideways deflection rather than a speed-up along the path.
The magnetic force on a charge is described by F = qvB sin(theta), where theta is the angle between the velocity and magnetic field. Its direction is found with the right-hand rule for a positive charge, and reversed for a negative charge. Because the force is perpendicular to the velocity, a uniform magnetic field can make a charged particle move in a circle with radius r = mv/(|q|B).
In a velocity selector, electric and magnetic forces cancel only for particles with speed v = E/B.
Key Facts
- Magnetic force magnitude: F = |q|vB sin(theta).
- Vector form: F = q v x B, so the force is perpendicular to both velocity and magnetic field.
- If v is parallel or antiparallel to B, then theta = 0 degrees or 180 degrees and F = 0.
- For circular motion in a uniform magnetic field: r = mv/(|q|B).
- Cyclotron period and frequency: T = 2πm/(|q|B) and f = |q|B/(2πm).
- Velocity selector condition: qE = qvB, so v = E/B when electric and magnetic forces balance.
Vocabulary
- Magnetic field
- A region where moving charges or magnetic materials can experience magnetic forces.
- Magnetic force
- The force on a moving charge caused by its motion through a magnetic field.
- Right-hand rule
- A method for finding the direction of magnetic force on a positive charge by pointing fingers along velocity and curling toward the magnetic field.
- Uniform circular motion
- Motion at constant speed around a circle caused by a force directed toward the center.
- Velocity selector
- A device that uses crossed electric and magnetic fields to allow only particles with a specific speed to pass straight through.
Common Mistakes to Avoid
- Using F = qvB for every angle, which is wrong because only the perpendicular part of velocity contributes. Use F = |q|vB sin(theta).
- Forgetting to reverse the direction for a negative charge, which gives the force direction for the wrong sign. Find the direction for a positive charge first, then flip it if q is negative.
- Thinking the magnetic force changes the particle's speed in a uniform field, which is wrong because the force is perpendicular to velocity. It changes direction, not kinetic energy.
- Confusing field direction symbols, which can reverse the answer. Dots mean magnetic field out of the page, while crosses mean magnetic field into the page.
Practice Questions
- 1 A proton with charge 1.60 x 10^-19 C moves at 3.0 x 10^6 m/s perpendicular to a 0.50 T magnetic field. What is the magnetic force magnitude?
- 2 An electron moves perpendicular to a 0.020 T magnetic field with speed 4.0 x 10^6 m/s. Using m_e = 9.11 x 10^-31 kg and |q_e| = 1.60 x 10^-19 C, find the radius of its circular path.
- 3 A positive charge moves to the right through a magnetic field directed into the page. What direction is the magnetic force, and how would the answer change if the charge were negative?