Magnetic force on charges and currents explains how moving charges, wires, and current loops interact with magnetic fields. Students need this cheat sheet to connect vector directions, formulas, and real physical motion in one place. It is especially useful for problems involving particles in fields, mass spectrometers, motors, and current-carrying wires.
The most important idea is that magnetic force acts only on moving charges or currents and depends on the component of motion perpendicular to the magnetic field. For a single charge, the force magnitude is , and its direction comes from the right-hand rule with charge sign included. In a uniform magnetic field, perpendicular motion can produce circular motion with .
For currents, the main formulas are for a straight wire and for a current loop.
Key Facts
- The magnetic force on a moving charge has magnitude , where is the angle between and .
- The vector form of magnetic force on a charge is , so a negative charge feels force opposite the right-hand-rule direction.
- A charge moving parallel or antiparallel to a magnetic field feels no magnetic force because and .
- A charge moving perpendicular to a uniform magnetic field follows circular motion when magnetic force supplies centripetal force, so .
- The radius of circular motion in a uniform magnetic field is .
- The angular speed and period for perpendicular circular motion are and .
- The magnetic force on a straight current-carrying wire is , where is the wire length inside the magnetic field.
- The torque on a current loop is , where is the number of turns and is the loop area.
Vocabulary
- Magnetic field
- A region described by where moving charges or currents can experience magnetic force.
- Lorentz force
- The total electromagnetic force on a charge, often written .
- Right-hand rule
- A direction rule where fingers follow or current, curl toward , and the thumb gives the force direction for positive charge or conventional current.
- Centripetal force
- The inward net force needed for circular motion, with magnitude .
- Conventional current
- The direction of current defined as the motion of positive charge, even though electrons in metals move the opposite way.
- Magnetic torque
- The turning effect on a current loop in a magnetic field, with magnitude .
Common Mistakes to Avoid
- Using for every angle is wrong because the correct magnitude is and only the perpendicular component matters.
- Forgetting the sign of the charge gives the wrong force direction because reverses direction for negative charges.
- Thinking magnetic force changes speed is wrong for a charge in a magnetic field alone because is perpendicular to and changes direction, not kinetic energy.
- Using the full wire length when only part is in the field is wrong because uses the length inside the magnetic field.
- Confusing the angle in torque problems is wrong because uses the angle between the loop’s area vector and , not necessarily the plane of the loop.
Practice Questions
- 1 A proton with charge moves at perpendicular to a magnetic field of . What is the magnetic force magnitude?
- 2 An electron moves perpendicular to a uniform magnetic field of with speed . Using and , find the radius of its circular path.
- 3 A straight wire of length carries current at to a magnetic field of . Find the force on the wire.
- 4 A charged particle enters a uniform magnetic field with velocity exactly parallel to . Explain why its path does not curve due to the magnetic field.
Understanding Magnetic Force on Charges & Currents
A magnetic field changes a particle’s direction without directly changing its speed. This happens because the push is sideways to the motion. A sideways force bends a path but does not add kinetic energy.
This is different from an electric field, which can speed up or slow down a charge. If a particle enters a field with one part of its velocity along the field and another part across it, the path becomes a helix. It moves forward along the field while circling around it.
This idea helps explain the spiral paths of charged particles trapped around Earth by its magnetic field. At very high speeds, the simple school formulas need relativistic corrections.
Direction is often the hardest part of these problems. Treat velocity, magnetic field, and force as three directions at right angles. For a positive charge, use the right hand rule in one consistent way.
Point your fingers in the direction of motion, curl them toward the magnetic field, and your thumb gives the force direction. Reverse that answer for an electron or any negative charge. Page diagrams use dots for a field coming out of the page and crosses for a field going into the page.
A useful habit is to draw a small coordinate sketch before choosing a direction. Check that the force is perpendicular to both the motion and the field.
The force on a wire comes from forces on many moving charge carriers inside it. Those carriers push on the metal lattice, so the whole wire can move. This is why a wire in a magnetic field can experience a noticeable force even though individual electrons have tiny masses.
Two nearby parallel wires provide an important example. Currents in the same direction attract, while currents in opposite directions repel. The result follows from the magnetic field made by each wire acting on charges in the other wire.
In calculations, use only the part of the wire that lies inside the relevant field. A bent wire may need to be split into straight sections because each section can feel a force in a different direction.
A current loop has forces on opposite sides, and those forces can form a turning effect rather than a simple sideways push. This is the basic action inside many electric motors. The loop tends to rotate toward an orientation where its magnetic effect lines up with the external field.
Once it reaches that orientation, the turning effect becomes zero. A motor keeps rotating by reversing the current at the right time with a commutator or electronic circuit. When studying torque, pay close attention to the angle given.
It is the angle between the loop’s area direction and the magnetic field, not usually the angle drawn along the wire itself. Units are another common source of errors. Convert milliamps to amps, centimeters to meters, and use tesla for magnetic field strength before calculating.