Moving Charges and Magnetism: common doubts, answered
The questions students ask most often about Moving Charges and Magnetism, each with a short answer. For the full chapter, read the Moving Charges and Magnetism notes.
About the chapter
How do I keep the direction rules in this chapter straight?
Use one tool throughout: the cross product with the right-hand rule. The force on a moving charge is along qv × B, the force on a wire along Il × B, and the field of a current element along dl × r. For a straight wire, grip it with the right hand, thumb along the current, and the fingers show B. Then reverse any force for a negative charge.
Magnetic field and the Lorentz force
Read this section in the notes →Why does a magnetic field do no work on a moving charge?
Because the magnetic force qv × B is always perpendicular to the velocity. A force at right angles to the motion changes only the direction, never the speed, so the kinetic energy stays constant. That is why a charge in a pure magnetic field moves in a circle or a helix at steady speed. To speed a charge up or slow it down you need an electric field.
How do I find the direction of the magnetic force on an electron?
Work out the direction of v × B as if the charge were positive, then reverse it for the electron. The force is qv × B, and q is negative for an electron, so its force points opposite to that on a proton moving the same way. Forgetting this flip is the commonest error in direction questions, including those asking which way a beam curves.
When is the magnetic force on a moving charge zero?
When the charge is at rest or moves parallel or antiparallel to the field. The size of the force is qvB sin θ, where θ is the angle between v and B, so it vanishes for θ = 0° or 180° and is largest at 90°. A neutral particle feels no magnetic force however fast it moves, and a charge at rest feels none however strong B is.
Force on a current-carrying conductor
Read this section in the notes →Where does the formula F = Il × B for a current-carrying wire come from?
It is the sum of the magnetic forces on the drifting charges inside the wire. Each carrier feels qv_d × B; a length l of wire with area A holds nAl carriers, and since I = neAv_d the total comes to F = Il × B, with l pointing along the current. The force is largest when the wire is perpendicular to B and zero when it lies along B.
Motion in a magnetic field
Read this section in the notes →Why does the time period of a charge circling in a magnetic field not depend on its speed?
Because a faster charge moves in a proportionally bigger circle. The radius is r = mv/(qB), so doubling v doubles r and the distance round the circle. The time for one turn, T = 2πr/v = 2πm/(qB), therefore has no v in it. The frequency ν = qB/(2πm) depends only on the charge-to-mass ratio and the field.
Why does a charged particle move in a helix in a magnetic field?
Because its velocity has a part along B and a part across B. The perpendicular part makes it go round in a circle of radius mv⊥/(qB), while the parallel part feels no force and carries it steadily along the field. Together they trace a helix. The distance it moves along B during one turn is the pitch, p = 2πmv∥/(qB).
Biot–Savart law
Read this section in the notes →What decides the direction of the magnetic field in the Biot–Savart law?
The field is along dl × r, perpendicular to both the current element and the line joining it to the point. For a straight wire, use the right-hand rule: thumb along the current, and the curled fingers show the field circling the wire. The field is zero at any point lying on the line of the current element itself, because sin θ = 0 there.
Field on the axis of a circular loop
Read this section in the notes →Why does the formula for the field at the centre of a circular loop have no π?
Because every element of the loop is at the same distance R from the centre and adds in the same direction. Adding μ₀I dl/(4πR²) over the whole circumference 2πR cancels the π, leaving B = μ₀I/(2R), or μ₀NI/(2R) for N turns. A semicircular arc gives half the full-loop value. The π remains only in the straight-wire result μ₀I/(2πr).
How does the field along the axis of a current loop change as you move away from its centre?
It falls with distance x as B = μ₀IR²/[2(x² + R²)^(3/2)]. At the centre, x = 0, this becomes μ₀I/(2R). Far away, where x is much larger than R, it falls as 1/x³, exactly like the axial field of a dipole. So a current loop seen from a distance behaves as a magnetic dipole whose moment is the current times the loop's area.
Ampere's circuital law
Read this section in the notes →What is the difference between the Biot–Savart law and Ampere's circuital law?
Biot–Savart gives the field of a small current element, which you add up over the conductor; Ampere's law links the line integral of B round a closed loop to the current passing through it. Biot–Savart works for any shape but needs integration. Ampere's law, ∮B·dl = μ₀I, gives B quickly only when symmetry makes B constant along a well-chosen loop, as for a long wire or solenoid.
What is the magnetic field inside a thick wire carrying a uniform current?
Inside, it grows in proportion to distance from the axis: B = μ₀Ir/(2πa²) for r < a, where a is the wire's radius. An Amperian circle of radius r inside encloses only the fraction r²/a² of the current. The field is zero on the axis, largest at the surface, where it equals μ₀I/(2πa), and outside it falls as 1/r like a thin wire's.
The solenoid
Read this section in the notes →Why is the magnetic field inside a long solenoid uniform?
Inside, the fields of all the closely wound turns add along the axis, while outside the contributions from opposite sides of each turn largely cancel. Applying Ampere's law to a rectangle with one side inside gives BL = μ₀nLI, so B = μ₀nI. Nothing in this result depends on where the side is placed inside, so the field is the same throughout the interior.
Is n in B = μ₀nI the total number of turns of the solenoid?
No, n is the number of turns per unit length, n = N/L. A solenoid with 500 turns over 0.5 m has n = 1000 per metre. If the winding has several layers, add the turns of every layer before dividing by the length. The field therefore depends on how tightly the coil is wound and on the current, not on the solenoid's radius.
Force between parallel currents
Read this section in the notes →Why do parallel currents attract when like charges repel?
Because each wire sits in the magnetic field of the other, and for currents in the same direction the force Il × B on each wire points towards the other. Working out the directions with the right-hand rule confirms it. So the rule runs opposite to electrostatics: parallel currents attract, antiparallel currents repel. The force per unit length is μ₀I₁I₂/(2πd).
Torque on a current loop
Read this section in the notes →When is the torque on a current-carrying coil in a magnetic field maximum?
When the plane of the coil is parallel to the field, so its normal is at 90° to B. The torque is τ = NIAB sin θ, where θ is measured between B and the normal to the coil, not the coil's plane. At θ = 90° it reaches NIAB. When the coil's plane is perpendicular to B, θ = 0 and the torque is zero, the stable position.
Why is the net force on a current loop zero in a uniform magnetic field?
The forces on opposite sides of the loop are equal and opposite, because the current runs in opposite directions along those sides, so they add to zero. Some of these pairs, however, do not act along the same line, and they form a couple that turns the loop. A uniform field thus produces torque but no net force; a non-uniform field can produce both.
Current loop as a magnetic dipole
Read this section in the notes →In what way is a current loop like a bar magnet?
A current loop has a magnetic moment m = IA, directed along its normal by the right-hand rule, and its far field has the same form as a bar magnet's. On the axis, B = (μ₀/4π)(2m/x³), and in the plane of the loop the field has size (μ₀/4π)(m/x³), pointing opposite to m, for x much larger than the radius. The face on which the current appears anticlockwise behaves as a north pole.
The moving coil galvanometer
Read this section in the notes →Why is the magnetic field in a moving coil galvanometer made radial?
So that the plane of the coil always lies along the field, whatever the deflection. The torque is then always NIAB, with sin θ = 1, and it is balanced by the spring's restoring torque kφ. This gives φ = (NAB/k)I, a deflection directly proportional to the current, so the scale can be marked in equal steps. A soft iron core inside the coil makes the field radial and stronger.
How is a galvanometer converted into an ammeter?
By connecting a small resistance, called a shunt, in parallel with it. Most of the current then bypasses the coil, and only the full-scale current I_G flows through the galvanometer. Because the shunt and the galvanometer share the same voltage, rₛ = I_G R_G/(I − I_G). The ammeter's overall resistance becomes very small, so placing it in series hardly changes the current it measures.
How is a galvanometer converted into a voltmeter?
By connecting a large resistance in series with it. The resistance is chosen so that full-scale current flows when the whole combination has the desired range voltage across it: R = V/I_G − R_G. The high total resistance means the voltmeter, connected in parallel with a component, draws very little current and so barely disturbs the voltage it is measuring.
Does increasing the number of turns always make a galvanometer more sensitive?
It raises current sensitivity but need not raise voltage sensitivity. Current sensitivity, φ/I = NAB/k, grows in proportion to N. But more turns also mean a longer wire, so the coil's resistance grows too, and voltage sensitivity, φ/V = NAB/(kR), can stay the same. A meter that responds to tiny currents is not automatically better at detecting tiny voltages.
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