Unit notes

A moving charge in magnetic field B experiences F=qv×B. The magnitude is |q|vB sinθ. Use the right-hand rule for positive charge, then reverse for negative charge. A stationary charge has no magnetic force from B alone.

Because magnetic force is perpendicular to velocity, it does no work on a point charge. In a uniform field, the perpendicular velocity component produces circular motion with radius r=mv_perp(|q|B). A parallel component remains constant, creating a helix. Magnetic forces can change momentum without changing speed.

For mutually perpendicular E, B, and velocity, electric and magnetic forces can oppose. An undeflected particle then satisfies |q|E=|q|vB, giving v=EB. The cancellation direction must be checked; the magnitude equation alone does not determine which travel direction works.

A downstream magnetic region can separate particles by mass-to-charge ratio. In the nonrelativistic model, r=mv(|q|B). If particles instead receive the same accelerating voltage, use |q|ΔV=mv²2 first: equal voltage does not mean equal speed for different masses.

The Biot–Savart law gives dB=(μ₀4π)I dl×r̂/r² for a steady current element. Both the direction and distance vary with geometry. On the axis of a circular loop, transverse contributions cancel; at its center B=μ₀I(2R) for one turn.

A long straight wire produces B=μ₀I(2πr), tangent to circles around the wire. Choose the direction using the right-hand grip rule. Superpose fields from multiple wires as vectors; two parallel currents in the same direction attract, while opposite currents repel.

For steady currents, ∮B·dl=μ₀I_enclosed. A circular path around a long straight wire works because the field is tangent with constant magnitude. Inside a uniform cylindrical current, enclosed current grows as r², giving B proportional to r; outside, B decreases as 1r.

Inside a sufficiently long ideal solenoid B≈μ₀nI and the exterior field is negligible away from ends. A rectangular Amperian path exploits that model. A short coil still obeys the law, but end effects prevent simply treating B as uniform everywhere.

For a uniformly filled wire of radius R, the field is zero at its center, rises linearly inside, and joins the inverse-radius outside solution continuously. The graph shape differs from the inverse-square exterior electric field of a charged sphere.

If current density is nonuniform, integrate J over the enclosed cross-section before applying Ampère’s law. The total current does not automatically belong inside a path of radius smaller than the wire. Always state the current direction when reporting a magnetic field direction.

The magnetic force on a moving charge is qv×B. Its magnitude is |q|vB sin θ, with θ between velocity and field. A stationary charge has no magnetic force from this term, and motion parallel to the field also gives zero force. Reverse the direction obtained for a positive charge when q is negative.

The magnetic force is perpendicular to velocity and therefore does no work on an isolated point charge through that force alone. It can change direction without changing speed. In a uniform field, a perpendicular velocity component can produce circular motion while a parallel component persists, producing a helical path.

For circular motion, equate magnetic-force magnitude to mv²r. The resulting radius depends on momentum magnitude divided by |q|B. Do not confuse this orbital relation with a statement that all particles in a beam have the same radius regardless of mass, speed or charge.

A current-carrying wire segment in a field experiences a force determined by current direction, segment orientation and B. A current loop can have zero net force in a uniform field but a nonzero torque. Its magnetic moment is associated with current, area and loop orientation.

The torque tends to align the magnetic moment with the field. The angle is between the moment vector, normal to the loop, and B—not necessarily between the plane of the loop and B. Draw the area normal to avoid a sine/cosine swap.

Two parallel currents exert forces through each other’s magnetic fields. Use the field of one wire and the force on the other rather than adding an invented attraction rule without direction. Parallel currents in the same direction attract in the standard long-wire geometry; opposite currents repel.

Biot–Savart builds a field from current elements, with direction set by a cross product and magnitude depending on source distance and geometry. As with electric-field integration, symmetry can cancel components. A current element is not an isolated point charge, so the electric inverse-square formula cannot simply be reused.

Ampère’s law relates circulation of B around a closed path to enclosed current in the magnetostatic setting. To extract a field magnitude easily, use a symmetry that makes B tangent and constant along suitable parts of the path. A long straight wire supports a circular path; an ideal long solenoid supports an appropriate rectangular path.

A zero enclosed current gives zero net circulation, not necessarily zero field everywhere along the path. External currents can contribute locally. Inside a current-carrying wire with a stated uniform current density, enclosed current depends on the fraction of cross-sectional area inside the chosen loop.

Outside a long straight wire, field magnitude decreases as 1r. Inside a uniformly conducting cylindrical wire with uniform current density, enclosed current grows as r², so the field grows proportional to r. These are different regions with different enclosed-current expressions.

The ideal long-solenoid interior field is approximately uniform away from ends, while edge fields require a more detailed model. A photographed electromagnet is not automatically an ideal solenoid; coils, cores and gaps can alter the field distribution.

For every direction problem, identify whether the requested arrow is current, field, force or magnetic moment. A correct right-hand rule applied to the wrong pair of vectors still gives the wrong answer. Label each vector before using the rule.

For a particle, F=qv×B is perpendicular to velocity, so its instantaneous power F·v is zero. A static magnetic field alone can bend a trajectory without changing speed. Electric fields can change kinetic energy, so do not extend this statement to a region containing both fields.

For velocity perpendicular to a uniform field, equate |q|vB to mv²r to obtain r=mv(|q|B). The sign of charge changes the bending direction, not this positive radius. A component parallel to the field remains unchanged in the ideal model, producing helical motion when combined with circular perpendicular motion.

The magnetic force on a current segment is I dℓ×B, integrated when needed. A loop can experience torque even when its net force is zero. The magnetic dipole moment points according to the current’s right-hand rule, and torque tends to align it with the field. Distinguish net translation from rotation.

Biot–Savart builds a field from current elements, including direction through a cross product and distance dependence. It is suited to geometries such as a finite arc or a ring when their contributions can be integrated. Source-element direction and observation-point geometry both matter.

Ampère’s law relates circulation of B around a closed path to enclosed current in the magnetostatic situations studied here. To solve easily for B, justify constant magnitude and alignment along the chosen path, as for an ideal long straight wire or long solenoid. Choosing a circular path around an arbitrary circuit does not create circular symmetry.

For a long solenoid, the ideal internal field is approximately μ₀nI and the external field is small away from ends. A real finite solenoid has fringe fields. State the approximation and avoid interpreting an ideal diagram as proof that the outside field is exactly zero everywhere.