Magnetic Field — Sources, Properties & Right-Hand Rule

A magnetic field describes how magnets, moving charges, and electric currents influence other moving charges and magnetic materials in the space around them. It is written as $B$ and measured in tesla $(\mathrm{T})$. The main idea is directional: a magnetic field is a vector field, so at each point it has both a size and a direction, and the right-hand rule is the usual shortcut for finding that direction in simple cases.

The Force Law And Its Symbols

You can think of a magnetic field as the part of the electromagnetic environment that tells you how a moving charge or a current-carrying wire would be pushed. For a charge $q$ moving with speed $v$ at angle $\theta$ to the field, the magnetic-force magnitude is

This condition matters. If the charge is not moving, the magnetic part of the force is zero. If it moves exactly parallel or antiparallel to the field, then $\sin\theta = 0$ and the magnetic force is also zero.

A common special-case result for the field around a long straight wire carrying steady current, at distance $r$, in vacuum or air, is

where $\mu_0$ is the permeability of free space.

Why The Field Falls Off As 1/r

The $1/r$ dependence in $B = \mu_0 I/(2\pi r)$ is not arbitrary; it follows from symmetry. By the right-hand rule, the field circles the wire, so on a circle of radius $r$ centered on the wire the field has the same magnitude all the way around and points along the circle. Summing the field contribution around that loop gives a quantity proportional to $B$ times the circumference $2\pi r$, and that total is fixed by the enclosed current $I$ (times $\mu_0$). Setting $B \cdot 2\pi r = \mu_0 I$ and solving gives $B = \mu_0 I/(2\pi r)$ directly. So the field weakens with distance precisely because the same "amount of circulation" is spread around an ever-larger circle. The same reasoning explains why the formula is valid only for a long straight wire: it relies on the circular symmetry that other wire shapes break.

Key Properties To Remember

Magnetic field is a vector, so direction is part of the answer, not an extra detail. Magnetic fields add by superposition: if two sources create fields at the same point, the net field is the vector sum. Field lines are a visual aid, not physical strings; at any point the field direction is tangent to the field line. In standard introductory treatment, magnetic field lines form closed loops rather than starting and ending the way electric field lines do on charges.

The Right-Hand Rule For A Straight Current

For a straight wire carrying conventional current, point your right thumb in the direction of the current; your curled fingers show the direction of the magnetic field circling the wire. This is one of the most useful versions of the rule because it gives direction quickly without extra algebra. Be careful about the current definition: the rule uses conventional current, which points the way positive charge would move. In a metal wire, electrons drift the opposite way.

Worked Example: Field Around A Long Straight Wire

Suppose a long straight wire carries a steady current upward, and you want the field at a point to the right of the wire. Use the right-hand rule: point your right thumb upward with the current, and your fingers curl around the wire. At the point to the right, the field points into the page.

For the strength, use the long-straight-wire formula. If $I = 5.0\ \mathrm{A}$ and $r = 0.020\ \mathrm{m}$,

So the field has magnitude $5.0 \times 10^{-5}\ \mathrm{T}$ and direction into the page. That example shows the two parts of a magnetic-field answer: magnitude from the formula, direction from the right-hand rule.

Practice Check

Try the same wire, but place the point to the left instead of the right, keeping the same current and distance. First use the right-hand rule to get the direction (it should flip to out of the page), then confirm the magnitude is unchanged because $r$ is the same.

Common Mistakes

• Treating magnetic field as a scalar and giving only the size.
• Forgetting that the right-hand rule uses conventional current, not electron flow.
• Using $B = \mu_0 I / (2\pi r)$ for any shape of wire, even when the wire is not approximately long and straight.
• Assuming a magnetic field always pushes a charge. A stationary charge feels no magnetic force.
• Mixing up the direction of the field with the direction of the force on a moving charge.

Where The Concept Is Used

Magnetic fields are used in motors, generators, transformers, MRI systems, speakers, compasses, and charged-particle motion. They also sit behind many circuit and electromagnetism ideas: once current creates a magnetic field, you can explain inductors, electromagnets, and why changing fields matter in electromagnetic induction.