The Motor Effect

From moving charges to current-carrying wires

Hang a horizontal copper wire between the poles of a horseshoe magnet and connect it to a battery. The wire jumps sideways — pushed by a force you cannot see, generated by the interaction between the current and the magnetic field. This is the motor effect: a current-carrying conductor in a magnetic field experiences a measurable mechanical force. Billions of electrons drifting along the wire each feel a tiny F = qvB push; summed across all the charge carriers, the total force on the wire is:

If a wire of length $l$ carries current $I$, the total charge passing through in time $t$ is $Q = It$. These charges move at average drift velocity $v_d$, so $l = v_d t$. The force on all moving charges in the wire becomes:

Motor effect force: $F = BIl\sin\theta$ (N). Maximum $F = BIl$ when $\theta = 90°$; zero when $\theta = 0°$ (wire parallel to B). Direction: right-hand palm rule — thumb = current $I$, fingers = $\vec{B}$, palm push = $\vec{F}$. Derived from $F = qvB$ using $Q = It$ and $l = v_d t$.

Pause — copy the highlighted formula and right-hand rule into your book before moving on.

What controls the motor effect force

We just saw that $F = BIl\sin\theta$ describes the motor effect force. That raises a question: which variables can we practically control to increase or decrease the force in a real device? This card answers it → $B$, $I$, $l$, and $\theta$ all contribute; multi-turn coils effectively multiply $l$ by the number of turns.

The motor effect force depends on four variables. Understanding how each one affects the force is essential for both problem-solving and real-world design.

• Magnetic field strength (B): Stronger magnets produce larger forces. High-performance motors use rare-earth magnets.
• Current (I): More current means more charge carriers moving, so more force. But higher current means more heating ($P = I^2R$).
• Wire length (l): Longer wires experience more force. In motors, this is achieved by coiling the wire into many turns.
• Angle ($\theta$): Maximum force at 90°, zero at 0°. The $\sin\theta$ dependence means small angles near 90° don't change the force much, but near 0° the force drops rapidly.

Applications of the motor effect:

• Loudspeakers: A coil attached to a paper cone sits in a permanent magnet's field. Audio current varies in direction and magnitude, making the cone vibrate and produce sound waves.
• Galvanometers: A coil in a magnetic field deflects proportionally to current. Used in analog meters.
• Electromagnetic pumps: Used to pump liquid metals (like sodium coolant in nuclear reactors) without moving parts.
• Maglev trains: Electromagnets create lift and propulsion forces using variations of the motor effect.

$F \propto B$, $I$, $l$, $\sin\theta$. Multi-turn coil: effective length $l_\text{total} = N \times \text{circumference}$. Loudspeaker: radial field keeps $\theta = 90°$ always; AC reverses current → force reverses → cone vibrates. Galvanometer: deflection $\theta \propto I$.

Add the highlighted factors and device applications to your notes before the check below.

Calculating force on a voice coil

We just saw how B, I, l, and θ each affect the motor effect force. That raises a question: in a real loudspeaker with 80 turns of wire, how do we compute the total force? This card answers it → multiply circumference by turn count to get total effective length, then apply $F = BIl$.

Real-world motor effect problems often involve multi-turn coils. The trick is to find the total effective length of wire in the field, then apply $F = BIl\sin\theta$.

For an $N$-turn coil: $l_\text{total} = N \times \pi d$. In a radial field $\theta = 90°$ at every point → $\sin\theta = 1$ always. Reversing current reverses force direction but keeps magnitude the same. Step order: circumference → total length → $F = BIl$.

Pause — write the highlighted coil worked-example method into your book before moving on.

Misconceptions — final check

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