Physics 2 · Unit 5: Magnetism and Electromagnetic Induction · 14 min read · Updated 2026-05-11
Force on Current-Carrying Wire in Magnetic Field — AP Physics 2
AP Physics 2 · Unit 5: Magnetism and Electromagnetic Induction · 14 min read
1. Core Origin of Force on Current-Carrying Wires★★☆☆☆⏱ 3 min
When current flows through a wire, it consists of countless moving charged electrons, each of which experiences a Lorentz force when placed in an external magnetic field. The net sum of these individual forces on all charges equals the total force acting on the wire as a whole.
This topic is a core component of Unit 5, accounting for 17–23% of the total AP Physics 2 exam score, and appears regularly in both multiple-choice and free-response sections, often paired with electric circuits or Newtonian mechanics.
2. Magnitude and Direction: Straight Wires in Uniform Fields★★☆☆☆⏱ 4 min
The force formula is derived from summing Lorentz forces on individual charge carriers. Substituting the definition of current $I = nA q v_d$ gives the general result:
F = BIL\sin\theta
Where $B$ is magnetic field strength, $I$ is current, $L$ is wire length, and $\theta$ is the angle between the current direction and magnetic field vector. If the wire is parallel to the field, force is zero; if perpendicular, force is maximized at $F = BIL$.
The vector form of the rule is:
\vec{F} = I \vec{L} \times \vec{B}
$\vec{L}$ is a vector pointing in the direction of current. Use the right-hand rule: point fingers along $\vec{L}$, curl toward $\vec{B}$, thumb points to the direction of $\vec{F}$.
Exam tip: Always measure $\theta$ between the current direction and the magnetic field vector, not between the force and the field. Double-check geometry before plugging in values.
3. Force Between Two Parallel Current-Carrying Wires★★★☆☆⏱ 3 min
Two parallel current-carrying wires exert force on each other: one wire produces a magnetic field that acts on the current in the second wire, and vice versa.
For two long parallel wires separated by distance $r$, carrying currents $I_1$ and $I_2$, the magnetic field from $I_1$ at $I_2$ is $B = \frac{\mu_0 I_1}{2\pi r}$. This field is always perpendicular to $I_2$, so $\sin\theta = 1$, giving force per unit length:
\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}
Direction rule: currents in the same direction attract, opposite directions repel. This rule is the basis for the formal definition of the ampere.
Exam tip: Don't rely solely on the 'same attract, opposite repel' memory rule. Re-derive direction with the right-hand rule to avoid mistakes.
4. Net Force on Curved Wires and Closed Loops★★★☆☆⏱ 3 min
For any curved wire in a uniform magnetic field, the net force equals the force that would act on a straight wire connecting the two endpoints of the curved wire, carrying the same current. This comes from integrating force over all small segments, where the integral of $d\vec{l}$ equals the net displacement between endpoints.
For a closed loop (start and endpoint are the same), net displacement is zero, so net force on the entire loop in a uniform field is always zero. This rule only applies to uniform magnetic fields.
Exam tip: Do not use the total length of the curved wire for this calculation. Net force depends only on endpoint separation for uniform fields.
5. AP-Style Concept Check★★★★☆⏱ 4 min
Common Pitfalls
Why: Students confuse this formula with magnetic flux (which uses $\cos\theta$) or misidentify the angle between vectors
Why: Students assume total length is always used, forgetting the endpoint rule for uniform fields
Why: Students rely on memory instead of confirming with first principles
Why: Students calculate force on individual segments but forget to add opposing force vectors
Why: Students confuse the source of the force, which always comes from an external field