Moving Charges and Magnetism

The One Thing

Moving electric charges produce magnetic fields, while magnetic fields exert forces on moving charges and current-carrying conductors. These principles, quantified by the Biot–Savart law, Ampere’s circuital law and the Lorentz force, explain the operation of galvanometers, ammeters, voltmeters, motors and cyclotrons.

Reactions, Processes and Experiments

Key Terms

• Magnetic field: The region around a magnet or moving charge where a magnetic force can be experienced. It is represented by the vector B and measured in tesla (T).
• Magnetic field lines: Imaginary lines whose tangent gives the direction of the magnetic field. They form closed curves and never intersect.
• Biot–Savart law: The law giving the magnetic field produced by a small current element: dB = (μ0/4π) I dl sinθ/r².
• Ampere’s circuital law: The line integral of magnetic field around a closed path equals μ0 times the enclosed current: ∮B·dl = μ0 I_enclosed.
• Lorentz force: The force on a charge q moving with velocity v in electric field E and magnetic field B: F = q(E + v × B).
• Magnetic force on a charge: In a magnetic field alone, the force is F = qvB sinθ.
• Force on a current-carrying conductor: A conductor carrying current I in a magnetic field experiences force F = I L × B, with magnitude F = BIL sinθ.
• Cyclotron motion: The circular motion of a charged particle entering a uniform magnetic field perpendicular to its velocity, with r = mv/(qB), ω = qB/m and T = 2πm/(qB).
• Magnetic dipole: A current-carrying loop behaving like a magnetic dipole with magnetic moment m = IA n̂.
• Magnetic moment: For a current loop, m = IA; its SI unit is A m².
• Torque on a current loop: The torque experienced by a current loop in a magnetic field is τ = m × B, with magnitude τ = IAB sinθ.
• Moving-coil galvanometer: An instrument that detects and measures small currents using torque on a current-carrying coil in a magnetic field.
• Current sensitivity: The deflection produced per unit current in a galvanometer: S_i = θ/I = NBA/C.
• Conversion of galvanometer: A galvanometer is converted into an ammeter by connecting a low resistance shunt in parallel, or into a voltmeter by connecting a high resistance in series.
• Tesla: The SI unit of magnetic field, where 1 T = 1 N A⁻¹ m⁻¹.
• Permeability of free space: μ0 = 4π × 10⁻⁷ T m A⁻¹.
• Right-hand thumb rule: A rule used to determine the direction of the magnetic field around a current-carrying wire.
• Deflecting torque: The torque acting on a galvanometer coil due to the magnetic field: τd = NIAB.
• Restoring torque: The mechanical torque opposing galvanometer deflection: τr = Cθ.
• Torsional constant: The constant C relating the restoring torque to the angular deflection in a galvanometer.
• Elementary charge: e = 1.602176634 × 10⁻¹⁹ C, used in the modern SI definition of the ampere.

Easily Confused

• Biot–Savart law and Ampere’s circuital law: Biot–Savart law applies generally to current elements, whereas Ampere’s law is especially convenient when the magnetic field has sufficient symmetry.
• Magnetic force and electric force: Magnetic force depends on velocity and is zero when velocity is parallel to the field; the Lorentz force includes both electric and magnetic contributions.
• Magnetic field direction for positive and negative charges: The right-hand rule gives the direction for positive charges; the force on a negative charge is opposite.
• Ammeter and voltmeter conversion: An ammeter uses a low-resistance shunt in parallel, whereas a voltmeter uses a high resistance in series.
• Deflecting torque and restoring torque: Deflecting torque is produced by the magnetic field, while restoring torque is mechanical and opposes deflection.
• Magnetic force and work: Magnetic force changes the direction of motion but does no work, so it does not change the speed or kinetic energy of an isolated charged particle.
• Magnetic dipole moment and magnetic field: The current loop has magnetic moment m = IA, whereas the magnetic field is the field produced by the loop or another source.

What Gets Asked

• Calculate the magnetic field due to a long straight wire, circular coil, solenoid or toroid; marks are lost by using the wrong geometry or omitting the factor μ0.
• Apply the Biot–Savart law or Ampere’s circuital law; marks are lost by using Ampere’s law without the required field symmetry or by failing to identify the enclosed current.
• Determine the path, radius, angular frequency or time period of a charged particle in a uniform magnetic field; marks are lost by treating the magnetic force as work-producing or by using a velocity component parallel to the field incorrectly.
• Analyse crossed electric and magnetic fields; the key condition is qE = qvB and v = E/B for undeflected motion.
• Find the force between parallel current-carrying wires; marks are lost by reversing the conclusions that parallel currents attract and antiparallel currents repel.
• Determine the torque, equilibrium deflection or current sensitivity of a galvanometer; marks are lost by confusing τd = NIAB with τr = Cθ or by using S_i = θ/I = NBA/C incorrectly.
• Convert a galvanometer into an ammeter or voltmeter; marks are lost by placing the shunt in series instead of parallel, or the high resistance in parallel instead of series, and by using the wrong resistance formula.