3.5. The force experienced by a current-carrying conductor in a magnetic field

The Motor Effect and Magnetic Flux Density

When a current-carrying conductor is placed inside an external magnetic field, the field around the conductor interacts with the external field. This produces a mechanical force on the conductor, known as the motor effect.

The force is greatest when the wire is perpendicular to the magnetic field lines. If the wire runs parallel to the field lines, the force is zero.

Fleming's Left-Hand Rule

Hold your left hand with your thumb, first finger, and second finger mutually at right angles:

• First finger = Field ($N \to S$)
• seCond finger = Current (conventional current, $+ \to -$)
• thuMb = Motion (direction of the resulting force)

Worked direction example: A horizontal wire carries current from left to right across a magnetic field directed vertically downwards (North pole above, South pole below). Point your first finger down and your second finger to the right. Your thumb points into the page, which is the direction of the force.

Defining Magnetic Flux Density ($B$)

Magnetic flux density ($B$) at a point in a magnetic field is a vector quantity whose magnitude equals the force experienced per unit length by a straight conductor carrying unit current placed perpendicular to the field at that point:

When a conductor of length $l$ carrying current $I$ sits at right angles to a uniform magnetic field $B$, the force is:

The SI unit of magnetic flux density is the tesla (T). The magnetic flux density is 1 tesla if a conductor of length $1\text{ m}$ carrying a current of $1\text{ A}$ at right angles to the field experiences a force of $1\text{ N}$.

Using unit analysis ($F = ma$, so $1\text{ N} = 1\text{ kg m s}^{-2}$):

Investigating the Force on a Current-Carrying Conductor (Higher Level)

This investigation examines how the electromagnetic force on a conductor relates to current and magnetic field.

Method and Variables

1. Place a strong U-shaped magnet assembly on a digital top-pan balance and tare the reading to zero.
2. Clamp a rigid horizontal wire loop from an external retort stand so that a section of length $l$ passes through the magnetic field between the poles without touching them.
3. Connect the wire in series with a DC power supply, a rheostat, an ammeter, and a switch.
4. Close the switch and record the current $I$. By Newton's third law, if the magnetic field exerts an upward force on the wire, the wire exerts an equal downward reaction force on the magnet assembly. The balance reading changes by $\Delta m$ (up or down, depending on current direction).
5. Calculate the electromagnetic force using $F = \Delta m \cdot g$, where $\Delta m$ is converted from grams to kilograms ($1\text{ g} = 10^{-3}\text{ kg}$) and $g = 9.8\text{ m s}^{-2}$.

Investigating the Relationships

• Force versus current ($F \propto I$): Keep the magnetic field $B$ and conductor length $l$ constant. Vary the current $I$ using the rheostat. A plot of $F$ on the y-axis against $I$ on the x-axis gives a straight line through the origin, confirming $F \propto I$. The slope of this line equals $Bl$, so the magnetic flux density is $B = \frac{\text{slope}}{l}$.
• Force versus magnetic field ($F \propto B$): Keep current $I$ and length $l$ constant. Vary $B$ by adding matched ceramic slab magnets to the yoke or using magnets of known flux density (measured using a Hall probe). Plotting $F$ against $B$ gives a straight line through the origin, confirming $F \propto B$.

Precautions and Sources of Error

• Precautions: Zero (tare) the balance before each run; verify that the wire does not touch the magnets; switch off current between readings so the wire does not heat up; read the balance quickly and steadily.
• Sources of error: Limited balance resolution for small force values; wire heating, which changes resistance and causes current to drift; edge effects (the magnetic field not being entirely uniform at the magnet ends); uncertainty in measuring the exact length $l$ of wire inside the field.

The Direct Current (DC) Motor

A direct current motor converts electrical energy into rotational kinetic energy using the motor effect. Some energy is inevitably lost as heat in the copper coils ($I^2Rt$) and as friction at the brushes and axle bearings.

Visual Arrangement and Drawing Guide

To draw a simple DC motor:

• Draw two magnetic poles facing each other: North on the left and South on the right, providing a horizontal magnetic field from left to right.
• Place a rectangular armature coil horizontally between the poles on a central axle.
• Connect the ends of the coil to a split-ring commutator on the axle: two semi-cylindrical copper segments separated by a small insulating gap.
• Draw two carbon (graphite) brushes pressing lightly against the outside of the split ring, connected to a DC battery.
• Label the current directions: into the page on one side of the coil, out of the page on the other.

Principle of Operation and Couple

Current travels in opposite directions along the two sides of the coil perpendicular to the field. By Fleming's left-hand rule, one side experiences an upward force and the opposite side experiences an equal downward force ($F = BIl$). These two forces are equal in size, opposite in direction, and act along different lines, so they form a couple: a pair of forces that turns an object without moving it along. The couple turns the coil about its axle.

Role of the Split-Ring Commutator

When the coil reaches the vertical position, its plane is perpendicular to the field lines. Here, the couple forces pull directly outwards in line with the coil, so the turning effect drops to zero. The coil's momentum (its inertia) carries it just past this vertical position.

At this instant, each half of the split ring breaks contact with one brush and reconnects with the opposite brush. This reverses the direction of the current in each side of the coil every half-turn ($180^{\circ}$). Because the current reverses relative to the fixed magnetic field, the upward and downward forces switch sides, sustaining continuous rotation in one direction rather than oscillating back and forth.

Factors Increasing Motor Speed and Turning Effect

From $F = BIl$, the turning effect and rotational speed of a DC motor can be increased by:

1. Increasing the current ($I$) in the coil.
2. Using stronger permanent magnets or electromagnets (increasing $B$).
3. Increasing the number of turns ($N$) on the coil, as each turn adds an extra pair of forces.
4. Increasing the area of the coil or length ($l$) of the sides in the field.
5. Winding the coil around a soft-iron armature core to concentrate the magnetic flux.

Moving Charges, Beam Deflection, and Parallel Conductors

An electric current is a flow of charge. Because a current-carrying wire experiences a force in a magnetic field, individual free moving charges must also experience a deflecting force.

Derivation of $F = qvB$

For a charge $q$ moving with speed $v$ at right angles to a uniform magnetic field $B$:

1. Start with the force on a conductor: $F = BIl$.
2. Electric current is charge per unit time: $I = \frac{q}{t}$, so $F = B\left(\frac{q}{t}\right)l = Bq\left(\frac{l}{t}\right)$.
3. In time $t$, the charge travels the length $l$, so $\frac{l}{t} = v$.
4. Therefore, the force on a single moving charge is:

When applying Fleming's left-hand rule to moving charges, remember that the second finger points along conventional current:

• For a positive charge (e.g. a proton), current points in the direction of velocity $v$.
• For a negative charge (e.g. an electron beam), conventional current points opposite to the electron velocity vector.

An electron beam entering a uniform magnetic field perpendicularly is deflected into a circular arc. Because the magnetic force acts at right angles to velocity at every instant, it does no work on the charge ($W = Fs\cos 90^{\circ} = 0$). The particle's speed and kinetic energy remain constant, but its direction changes continuously.

The magnetic force provides the required centripetal force:

Faster or heavier particles follow larger circular paths, while stronger fields or larger charges produce tighter curves.

Forces Between Parallel Conductors

Two long parallel wires carrying currents exert magnetic forces on each other because each sits inside the magnetic field produced by the other:

• Currents in the same direction attract.
• Currents in opposite directions repel.

Historically (before the 2019 SI redefinition), the ampere was defined using this mutual force: the constant current which, maintained in two straight parallel conductors of infinite length and negligible cross-section placed 1 metre apart in a vacuum, produces a force of $2 \times 10^{-7}\text{ N}$ per metre between them. Today, the ampere is defined by fixing the numerical value of the elementary charge $e$.