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.
Understanding Physics: Magnetic Force on a Moving Charge
The magnetic part is only one part of the full Lorentz force. An electric field pushes a charge whether it is still or moving. A magnetic field needs the charge to have velocity across the field.
This difference matters because electric forces can transfer energy to a particle. They can make it gain or lose kinetic energy. A magnetic force alone cannot do this in the usual classical picture.
Its push stays at right angles to the particle's instant motion. The particle changes direction continuously, while its speed and kinetic energy stay constant. In many machines, electric fields provide the energy and magnetic fields guide the path.
When a particle enters a uniform field exactly sideways, the magnetic force acts like the inward force needed for circular motion. A faster particle makes a wider circle. A particle with more mass makes a wider circle too.
A stronger magnetic field bends the same particle more tightly. If the velocity has one part across the field and one part along it, the result is a helix. The particle circles around magnetic field lines while moving forward along them.
This motion occurs in plasma, in auroras, and around planets with magnetic fields. The stated cyclotron frequency is independent of speed only when the particle is moving far below the speed of light. At very high speeds, relativity increases the effective inertia, so the orbit timing changes.
Mass spectrometers use this bending to separate ions. Ions first need a known speed or a known energy. They then enter a magnetic region, where their curved tracks reveal their mass to charge ratio.
Ions with different masses curve by different amounts even if they carry the same charge. This helps scientists identify atoms and molecules in a sample. In a velocity selector, crossed electric and magnetic fields remove particles with unwanted speeds.
Electric motors use the same underlying force in a different form. Charges moving through a current carrying wire feel magnetic forces, producing forces on sections of wire and turning a rotor.
The Hall effect is another useful example. Charges are pushed toward one side of a conductor, creating a measurable voltage that can be used to sense magnetic fields.
Direction is usually the hardest part of these problems. First draw the velocity and field as arrows. Use the right hand only for a positive charge.
Point your fingers along the velocity, curl them toward the magnetic field, and your thumb gives the force direction. Reverse that result for an electron or any negative ion. A field drawn as dots points out of the page, while crosses point into the page.
Keep the three directions separate because the force must be perpendicular to both the velocity and the field. Check the angle before calculating.
Only the part of velocity across the field causes bending, so a small sideways component gives a small force. Use tesla for magnetic field, coulomb for charge, metre per second for speed, and newton for force.
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?