Magnetic Field

Magnetic field fundamentals, magnetic flux lines, and Lorentz force on moving electric charges.

Conceptual context

Electromagnetism unifies electricity and magnetism under a single theory. Just as the electric field describes the influence of charges on space, the magnetic field $B$ describes the influence of currents and magnets on the surrounding space. The central question is: what quantity quantifies the ability of a current to deflect other moving charges that pass nearby?

Magnetic phenomena have been known since antiquity: natural magnets (lodestone) attract iron and compasses point north. However, the connection with electricity was not discovered until 1820, when Ørsted observed that an electric current deflected a nearby compass needle. That discovery triggered the entire theory of electromagnetism. This leaf belongs to the electromagnetism fundamentals block and is the magnetic complement of the electric field.

Understanding the magnetic field allows predicting the motion of charged particles in plasmas, designing electric motors, and understanding the physics of nuclear magnetic resonance.

🟢 Essential level

The magnetic field $B$ at a point in space is a vector quantity describing the ability of that point to exert a force on a moving charge. The fundamental difference from the electric field is that $B$ only acts on moving charges: a charge at rest experiences no magnetic force, even in a region where $B$ is very intense.

The most direct intuition is to imagine a conductor through which a current $I$ flows. That current modifies the surrounding space, creating a field $B$ that points circularly around the wire. If a charge $q$ moving through that space is placed there, it experiences a force $F_m$ perpendicular to its velocity $v$ and to the field $B$. That perpendicularity has a key consequence: the magnetic force never does work, because it is always perpendicular to the displacement.

The magnitude $B$ exists even if no test charge is present, just as the electric field exists regardless of whether charges are there to detect it. It is a property of space created by currents and magnets. A compass detects $B$ because its atoms act as tiny magnets that align with the field: this is the most everyday way to visualize $B$ without any calculation.

[!NOTE] The direction of $B$ around a conductor is determined by the right-hand rule: if the right thumb points along the current $I$ direction, the curled fingers indicate the circulation direction of $B$.

🔵 Formal level

Basic formulas

The force that $B$ exerts on a charge $q$ moving with velocity $v$ at angle θ to the field is:

Magnetic Lorentz force

This formula has three immediate implications: the force is maximum when $v$ is perpendicular to $B$ (so $\sin\theta$ equals 1), zero when $v$ is parallel or antiparallel to $B$ (so $\sin\theta$ equals 0), and always perpendicular to the plane formed by $v$ and $B$. It also explains why a magnetic field can bend a trajectory without increasing the kinetic energy of the particle.

Real student questions

Why does the magnetic force do no work?

Because $F_m$ is always perpendicular to $v$. Work is the force component along the displacement; if they are perpendicular, the dot product is zero at every instant.

What is the difference between $B$ and $F_m$?

$B$ is a property of space created by currents and magnets; it exists even if no moving charges are present. $F_m$ is the actual force on a specific charge $q$ moving with velocity $v$ in that field.

Why are $B$ field lines always closed?

Because magnetic monopoles do not exist: there are no free magnetic charges where lines can start or end. $B$ field lines always form closed loops, exiting the north pole and entering the south pole of any magnet.

Final synthesis

The magnetic field $B$ describes the magnetic influence of currents and magnets on space, acting only on moving charges. It never does work: it only deflects trajectories. Its 1/r dependence for the straight wire distinguishes it from the Coulomb electric field. Mastering $B$ is essential for understanding motors, generators, magnetic resonance, and particle physics.