For a long straight wire, magnetic field strength changes in a simple, predictable way: larger current produces a stronger field, while greater distance from the wire weakens it.

Core relationship

A long straight wire with a steady current produces a magnetic field around it.

A long, straight current-carrying wire produces magnetic field lines that form concentric circles centered on the wire. The arrows indicate the magnetic-field direction around the wire (the direction is tangential to the circles), reinforcing the circular symmetry assumed in the long-wire model. Source

For this idealized geometry, the magnetic field strength at a point depends on two quantities only: the current in the wire and the perpendicular distance from the wire. More current makes the field stronger, while greater distance makes it weaker. This relationship is central because it allows quick comparisons of field strength without needing a full derivation each time.

The emphasis on perpendicular distance matters. The relevant separation is the shortest distance from the point to the wire’s center line, not some other measured length.

This equation is the exact free-space form of the syllabus statement. It shows that field strength increases linearly with current and decreases inversely with perpendicular distance.

What the proportionalities mean

Direct proportionality to current

If the distance from the wire stays fixed, the field strength changes by the same factor as the current. Double the current, and the field doubles. Triple the current, and the field triples. Reduce the current by half, and the field is halved. This is called a direct proportionality because $B$ and $I$ change together in a linear way.

This is especially useful when only the current changes. In that case, the geometry stays the same, so the comparison is simple and mostly conceptual.

Inverse proportionality to distance

If the current stays fixed, the field strength changes inversely with distance from the wire. Double the perpendicular distance, and the field becomes half as large. Move three times farther away, and the field becomes one-third as large. Move to half the original distance, and the field doubles.

This is an inverse relationship, not an inverse-square relationship. That distinction is important. The field weakens with $1/r$, so it decreases more gradually than a quantity that follows $1/r^2$.

Interpreting distance correctly

Because the equation uses $r$, problems often test whether you can identify the correct distance. For a long straight wire, every point that is the same perpendicular distance from the axis has the same field magnitude. The wire’s axis is the imaginary line running through its center.

That means:

• measure from the wire’s center line, not from its outer edge unless the geometry makes that necessary

• use the shortest distance to the axis

• do not use a distance measured along the wire

• do not use the length of a circular path around the wire

If a point is farther from the wire, the field there is weaker even if the current has not changed. If two points are equally far from the axis, the field magnitudes are equal for the ideal long straight wire model.

Comparing two situations

A very efficient AP Physics method is to compare two magnetic fields by using ratios instead of repeatedly substituting constants. From the main equation,

$\dfrac{B_2}{B_1}=\dfrac{I_2}{I_1}\dfrac{r_1}{r_2}$

This form is helpful when both current and distance change at the same time. It keeps the physical meaning clear: increasing current strengthens the field, while increasing distance weakens it.

When using this ratio idea, match the variables carefully. A current increase appears as $I_2/I_1$, while a larger distance reduces the field, so the distance factor appears as $r_1/r_2$.

Graphs and physical trends

For a fixed distance, a graph of magnetic field strength vs. current is a straight line through the origin. That reflects the direct proportionality between $B$ and $I$.

For a fixed current, a graph of magnetic field strength vs. distance is not a straight line.

Graph of magnetic field magnitude $B$ versus radial distance $r$ from the wire’s axis. Outside the wire ($r>R$), the curve is labeled $B\propto 1/r$, matching the AP Physics long-wire relationship; the plot also highlights that the behavior differs inside the wire ($r<R$), where the field increases approximately linearly with $r$ for uniform current density. Source

It curves downward because the field follows $1/r$. If you instead graph magnetic field strength vs. $1/r$, you get a straight-line relationship.

These graph shapes help identify whether data fit the model of a long straight wire. They also help distinguish this relationship from inverse-square behavior.

Limits and common mistakes

The equation is based on an ideal long straight wire. It works best when the point of interest is not near the ends of the wire and when the wire is much longer than the distance to the observation point.

Common mistakes include:

• treating the distance dependence as $1/r^2$ instead of $1/r$

• forgetting that the comparison uses perpendicular distance

• changing both current and distance but accounting for only one of them

• assuming the field stays the same when the point moves farther away

• using the wire’s physical length in the equation, even though it does not appear in the long-wire model