Magnetic Fields of Currents

01 / UNDERSTAND & APPLY

1. Magnetic force bends motion without supplying kinetic energy

A charge q moving with velocity v in electric and magnetic fields experiences the Lorentz force F = q(E + v × B). The magnetic part has magnitude |q|vB sinθ and is perpendicular to both v and B. Use the right-hand rule for a positive charge, then reverse for a negative charge. A stationary charge experiences no magnetic force, although an electric force may act.

Because (v × B)·v = 0, the magnetic force alone does no work on a point charge and does not change its speed or kinetic energy. It can change momentum direction. This is different from saying a magnetic field can never be involved in energy transfer in a device: moving conductors, induced electric fields and mechanical agents require a complete system account.

When velocity is perpendicular to a uniform B, the magnetic force is centripetal. Equating |q|vB = mv²/r gives r = mv/(|q|B) for nonrelativistic motion. Angular frequency is |q|B/m and period 2πm/(|q|B), independent of speed within this approximation. The radius grows with speed even though the period does not.

When velocity has a component parallel to B, that component remains unchanged while the perpendicular component circles. The path is a helix with radius mv⊥/(|q|B) and pitch v∥T. If v is entirely parallel to B, there is no deflection. At relativistic speeds, momentum replaces mv and the simple constant-frequency result no longer holds.

02 / UNDERSTAND & APPLY

2. From individual carriers to a force on a conductor

For a straight wire segment carrying current I in uniform B, the force is F = I L × B, where L points along conventional current and has magnitude equal to the active segment length. This follows by summing qv × B over the moving carriers. For a curved wire, integrate I dl × B. The wire’s force is transmitted through interactions between carriers and the material.

In a uniform field, the net force on a complete closed current loop is zero because ∮dl = 0, but the loop can still experience a torque. In a nonuniform field, a loop may also feel a net force. Distinguish a whole loop from one selected side; opposing side forces can cancel translation while forming a couple.

Crossed electric and magnetic fields can select a speed. For a particle moving perpendicular to both fields, electric and magnetic forces cancel when their directions oppose and v = E/B. This condition is independent of charge magnitude and mass, but the particle direction and field orientations must actually produce opposite forces. Equal magnitudes with the same direction do not select an undeflected trajectory.

A magnetic analyser after a selector can separate particles by mass-to-charge ratio because r = mv/(|q|B). A cyclotron uses alternating electric fields to supply kinetic energy at gaps and magnetic fields to bend the path; the magnetic field alone does not accelerate the speed. This separation of roles is essential to explaining the device.

03 / UNDERSTAND & APPLY

3. Build magnetic fields with the Biot–Savart law

A steady current element contributes dB = (μ₀/4π) I dl × r̂/r² at an observation point in vacuum. The vector dl follows conventional current and r̂ points from the element to the observation point. The cross product gives both the sinθ factor and direction. Integrate over the current path and superpose fields from all sources.

For a long straight wire, integration gives B = μ₀I/(2πr), circling the wire by the right-hand rule. The inverse-distance dependence differs from a point charge’s inverse-square field because the source is extended. Near the ends of a finite wire the long-wire formula fails; for a finite segment at perpendicular distance a, B = μ₀I(sinθ₁ + sinθ₂)/(4πa) when the endpoint angles are measured from the perpendicular in the usual straddling geometry.

At the centre of a circular loop of radius R, all element fields point normal to the loop and add to B = μ₀I/(2R). For N identical closely spaced turns, multiply by N. A circular arc subtending angle φ in radians contributes μ₀Iφ/(4πR) at its centre. Radial leads contribute zero there because dl is parallel to the displacement vector.

On the axis of a circular loop at distance x from its centre, B = μ₀IR²/[2(R² + x²)^(3/2)] for one turn. At x = 0 it reduces to the centre field; far away it falls as 1/x³ and resembles a magnetic dipole. Symmetry cancels transverse contributions, just as in electric ring-field calculations.

04 / UNDERSTAND & APPLY

4. Symmetry turns Ampère’s law into a field calculation

For steady currents in vacuum, Ampère’s law states ∮B·dl = μ₀Ienclosed. Choose the contour orientation by a right-hand rule and use signed current piercing its surface. The law is always a circulation statement, but extracting B directly needs symmetry that fixes direction and constancy along useful parts of the contour.

For a long straight wire, a circular path centred on the wire has B tangent and constant, giving B2πr = μ₀I. Inside a long solid wire of radius R with uniform current density, enclosed current is I r²/R², so B = μ₀Ir/(2πR²). The field rises linearly inside and falls inversely outside. A nonuniform current density changes the interior result.

A long tightly wound solenoid has approximately uniform interior field B = μ₀nI and small exterior field away from the ends, where n is turns per unit length. This follows using a rectangular Amperian loop with one long side inside and one outside. A short solenoid has significant fringing and cannot be treated as an infinite one everywhere.

For an ideal toroid with N turns, B = μ₀NI/(2πr) within the winding region; it varies with distance r from the central axis. In the ideal continuous-winding model the field is zero in the central hole and outside. A real discrete winding has leakage fields. Magnetic cores modify the material response and require the appropriate permeability model.

05 / UNDERSTAND & APPLY

5. Current–current force is an interaction through fields

Two long parallel wires separated by d exert force per length μ₀I₁I₂/(2πd) in vacuum. To determine direction, first find the field of one wire at the other, then apply I L × B. Parallel currents attract and antiparallel currents repel. The force on each wire has equal magnitude, consistent with the interaction symmetry in the steady idealisation.

The result assumes wires long compared with separation and the region considered, with return paths sufficiently remote not to alter the calculation appreciably. A real circuit is closed; ignoring the return wire is an approximation, not a claim that current can begin and end along an isolated segment.

For multiple wires, add force or field vectors rather than pairwise magnitudes. At a point between parallel equal currents, the magnetic fields oppose and can cancel; outside the pair they reinforce. Opposite currents give the opposite pattern. Similar-looking electrostatic cancellation diagrams cannot be reused without applying the magnetic right-hand rule.

06 / UNDERSTAND & APPLY

6. A current loop behaves as a magnetic dipole

A planar current loop has magnetic moment μ = NIA n̂, with area normal chosen by curling the right-hand fingers with conventional current. In a uniform magnetic field its torque is τ = μ × B and potential energy U = −μ·B. The loop tends to align its magnetic moment with the field, even though its net force is zero.

The angle in τ = μB sinθ is between the area normal and B, not between the plane of the loop and B. Torque is zero when the normal is parallel or antiparallel to B, corresponding to the loop plane perpendicular to the field. The parallel moment is stable; the antiparallel moment is unstable under small rotational disturbances.

If a loop is reshaped while wire length and current remain fixed, its area and therefore magnetic moment may change. A circular loop encloses the largest planar area for a fixed perimeter. If the number of turns changes using the same wire, both area per turn and N must be tracked rather than multiplying by N while pretending the original radius remains fixed.

A charged particle circulating in a classical orbit also constitutes a current. For angular momentum L, the orbital magnetic moment is μ = qL/(2m), so an electron’s orbital magnetic moment is opposite its angular momentum. Intrinsic spin magnetic moments require quantum theory and are not simply literal rotation of a charged sphere.

07 / UNDERSTAND & APPLY

7. Dipole fields connect loops, bar magnets and compass motion

Far from a small current loop, its magnetic field has the dipole form. On its axis B = (μ₀/4π)(2μ/r³) in the moment direction; on its equatorial plane B = (μ₀/4π)(μ/r³) opposite the moment. These formulas require observation distance much larger than the source size. Near a loop or finite bar magnet, the full geometry matters.

A bar magnet’s dipole moment conventionally points from its south to north pole. Magnetic field lines go from north to south outside the magnet and continue through it, forming closed loops. The pole language is a useful description of a dipole, but cutting a magnet produces smaller dipoles rather than an isolated north pole in ordinary magnetostatics.

In a nonuniform external field, a magnetic dipole can experience net force as well as torque. For a fixed small dipole moment in a static field, F = ∇(μ·B) gives the corresponding force model. In a uniform field only the orientation-dependent torque remains. This distinguishes a compass aligning from a magnet being attracted into a stronger-field region.

If a magnetic dipole with moment of inertia Jrot is free to oscillate in a uniform B, torque is −μB sinθ. For small angles, sinθ ≈ θ, giving simple harmonic motion with period 2π√(Jrot/(μB)). This result needs small angular amplitude, negligible damping and a fixed moment magnitude; it is not a universal period for all rotations.

08 / UNDERSTAND & APPLY

8. Magnetic flux uses area orientation and closed-loop geometry

Magnetic flux through a surface is ΦB = ∫B·dA. For uniform B through a flat area it is BA cosθ, where θ is between B and the area normal. Its unit is weber, equal to tesla square metre. Rotating the surface changes flux even if field magnitude and area stay fixed.

The total magnetic flux through a closed surface is zero in classical electromagnetism: ∮B·dA = 0. Unlike electric Gauss’s law, there is no enclosed magnetic-charge term in the standard theory. Field can enter and leave the surface, but there is no net source or sink of B. Zero net closed flux does not mean zero magnetic field.

For a coil, distinguish flux through one turn from flux linkage NΦ when each turn links the same flux. Changing flux linkage is important in induction, but flux itself is a geometric field integral rather than an emf. A steady field through a fixed loop can have nonzero flux and zero induced emf.

When a surface bounds a closed wire loop, changing the chosen spanning surface does not change total magnetic flux through that loop in a divergence-free field, provided the boundary orientation is consistent. This is why the loop’s flux linkage is well-defined. The detailed induced-electric-field analysis is a separate induction problem.

09 / UNDERSTAND & APPLY

9. Original practice and direction checks

Use right-hand rules before arithmetic and explicitly reverse the magnetic force for negative charge. The following problems are independently authored.

• Magnetic force uses velocity and charge sign, not merely field direction.
• Uniform B bends a free charged-particle path without changing speed.
• Use symmetry and limits to choose Biot–Savart or Ampère.
• Loop moment follows conventional current and the right-hand area normal.
• Dipole far fields decay as inverse cube and require a distant observation point.
• Flux and flux linkage are not the same as induced emf.