Magnetic Fields and Forces

Having finished electric fields, we now turn to magnetism — the second of the three fundamental classical field theories. A magnetic field is produced by moving charges (currents) and permanent magnets, and it exerts forces on other moving charges and current-carrying wires. In this lesson we meet the key quantity magnetic flux density $B$B, the force on a current-carrying conductor $F = BIL\sin\theta$F=BILsinθ, Fleming's left-hand rule (often called the right-hand-slap rule in modern textbooks), and the field patterns around wires, coils and solenoids that the OCR mark scheme insists on.

Spec mapping (OCR H556 Module 6.3 — electromagnetism). This lesson covers the definition of magnetic flux density as the force per unit current per unit length on a wire, $B = F/(IL)$B=F/(IL); the unit tesla and its equivalent forms; the force law $F = BIL\sin\theta$F=BILsinθ for a wire at any angle to the field; Fleming's left-hand rule for direction; field patterns around a straight wire, a flat coil and a solenoid; and the qualitative force between two parallel current-carrying wires.

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What Is a Magnetic Field?

A magnetic field is a region of space in which a moving charge (or a magnet) experiences a force. Magnetic fields are produced by:

1. Permanent magnets — ferromagnetic materials with aligned atomic magnetic moments (iron, nickel, cobalt and certain rare-earth alloys).
2. Electric currents — moving charges always generate a magnetic field around them (Hans Christian Oersted's discovery, 1820).
3. Changing electric fields — through Maxwell's "displacement current" term, beyond the OCR spec but mentioned for completeness.

The field is drawn as field lines which:

• Run from the north pole to the south pole outside a magnet, and south to north inside (closing the loop).
• Never cross or end in free space — they form closed loops.
• Are closer together where the field is stronger; field strength is the line density.

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Magnetic Flux Density

The strength of a magnetic field at a point is characterised by the magnetic flux density, symbol $B$B. Its SI unit is the tesla (T), defined operationally by the force on a current-carrying conductor (see below). One useful re-expression of the unit is

$1\ \text{T} = 1\ \text{N A}^{-1}\,\text{m}^{-1} = 1\ \text{V s m}^{-2} = 1\ \text{kg s}^{-2}\,\text{A}^{-1}.$1 T=1 N A−1m−1=1 V s m−2=1 kg s−2A−1.

The form $1$1 T $= 1$=1 V s m$^{-2}$−2 — equivalent to $1$1 Wb m$^{-2}$−2 — is the one that resurfaces in the flux-and-induction lesson; the form $1$1 T $= 1$=1 N A$^{-1}$−1 m$^{-1}$−1 is the one used in this lesson's force law.

Typical magnetic field strengths:

Source	$B$B / T
Earth's surface field	$\sim 5 \times 10^{-5}$∼5×10−5
Fridge magnet	$\sim 0.005$∼0.005
Loudspeaker magnet	$\sim 0.1$∼0.1
Strong permanent magnet (NdFeB)	$\sim 1.5$∼1.5
MRI scanner	$1.5 - 7$1.5−7
Superconducting research magnet	$10 - 45$10−45

The tesla is a big unit. The Earth's field is only about $50$50 microtesla — five orders of magnitude smaller than a hospital MRI.

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Force on a Current-Carrying Conductor

Place a straight wire of length $L$L carrying a current $I$I in a uniform magnetic field of flux density $B$B. If the wire makes an angle $\theta$θ with the field, the force on the wire is

$F = BIL\sin\theta.$F=BILsinθ.

The $\sin\theta$sinθ matters. OCR expects you to include it whenever the wire is not perpendicular to the field. Special cases:

• $\theta = 90^{\circ}$θ=90∘ (wire perpendicular to field): $F = BIL$F=BIL, the maximum.
• $\theta = 0^{\circ}$θ=0∘ (wire parallel to field): $F = 0$F=0.
• $\theta = 30^{\circ}$θ=30∘: $F = \tfrac{1}{2}BIL$F=21​BIL.

The direction of the force is perpendicular to both the current and the field — it is given by Fleming's left-hand rule (also known as the right-hand-slap rule when applied to positive charge motion).

Fleming's Left-Hand Rule (and the right-hand-slap variant)

Use your left hand, with the first three fingers mutually at right angles:

• First finger — direction of the Field $\vec{B}$B.
• Second finger — direction of the Current $\vec{I}$I (conventional, i.e. positive).
• ThuMb — direction of the resulting Motion / force $\vec{F}$F.

Alternative formulation (right-hand-slap rule): with the right hand flat, fingers in the direction of $\vec{B}$B, thumb in the direction of $\vec{I}$I (positive current), the palm pushes in the direction of $\vec{F}$F on a positive current. For an electron beam ($I$I is opposite to electron velocity), swap hand or reverse the answer.

flowchart LR
    A[Positive current direction] --> B[Use left hand]
    C[Field direction known] --> D[First finger along B]
    E[Current along second finger] --> F[Thumb gives force direction]
    G[Electrons / negative carriers] --> H[Reverse the answer<br/>OR use right hand]

The mermaid above encapsulates the decision procedure: identify whether the carriers are positive or negative, then apply the rule.

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Worked Example 1 — Force on a Horizontal Wire

A horizontal wire $80$80 cm long carries a current of $3.0$3.0 A perpendicular to a horizontal magnetic field of flux density $0.15$0.15 T. Calculate the force on the wire.

$F = BIL\sin\theta = (0.15)(3.0)(0.80)\sin 90^{\circ} = (0.15)(3.0)(0.80)(1) = 0.36\ \text{N}.$F=BILsinθ=(0.15)(3.0)(0.80)sin90∘=(0.15)(3.0)(0.80)(1)=0.36 N.

The force is $0.36$0.36 N, directed vertically (up or down depending on the orientation of $B$B and $I$I; you'd use Fleming's left-hand rule to determine which).

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Worked Example 2 — Wire at an Angle

A $20$20 cm wire carries a current of $5.0$5.0 A through a magnetic field of $0.40$0.40 T, making an angle of $30^{\circ}$30∘ with the field. Find the force on the wire.

$F = BIL\sin\theta = (0.40)(5.0)(0.20)\sin 30^{\circ} = (0.40)(5.0)(0.20)(0.5) = 0.20\ \text{N}.$F=BILsinθ=(0.40)(5.0)(0.20)sin30∘=(0.40)(5.0)(0.20)(0.5)=0.20 N.

Compare with the perpendicular case: if the wire were at $90^{\circ}$90∘ to the field, the force would be $(0.40)(5.0)(0.20)(1) = 0.40$(0.40)(5.0)(0.20)(1)=0.40 N. At $30^{\circ}$30∘ it is exactly half, as $\sin 30^{\circ} = 0.5$sin30∘=0.5. The component of the wire perpendicular to the field is what matters; the parallel component carries current that experiences no force.

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Worked Example 3 — The Current Balance

A "current balance" is a delicate laboratory device used historically to define the ampere. A straight wire of length $10$10 cm lies on one pan of a sensitive balance, horizontally, perpendicular to a magnetic field of $0.80$0.80 T. When a current is switched on the reading on the balance changes by $2.0$2.0 g. What is the current?

Force from balance reading. The change in mass reading corresponds to a downward (or upward) extra force of

$F = mg = (2.0 \times 10^{-3})(9.81) \approx 1.96 \times 10^{-2}\ \text{N}.$F=mg=(2.0×10−3)(9.81)≈1.96×10−2 N.

Apply F = BIL. Rearranging for the current,

$I = \frac{F}{BL} = \frac{1.96 \times 10^{-2}}{(0.80)(0.10)} \approx 0.245\ \text{A} \approx 0.25\ \text{A}.$I=BLF​=(0.80)(0.10)1.96×10−2​≈0.245 A≈0.25 A.

This is the classical method used to standardise the ampere before the 2019 SI redefinition, in which the ampere was redefined in terms of the elementary charge $e$e and the second.

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Field Patterns from Currents

Oersted (1820) discovered that a compass needle near a current-carrying wire is deflected — a current produces a magnetic field around itself. Three field patterns are required by the OCR spec:

1. Straight wire — concentric circles around the wire. The right-hand grip rule gives the direction: grip the wire with the right hand, thumb along the current, fingers curl in the direction of the field. The field strength falls off as $1/r$1/r with distance from the wire.

2. Flat circular coil — a "donut" field pattern with the field inside the loop along the axis. The right-hand grip rule applied to one segment shows that the contributions from all segments add along the axis.

3. Solenoid — almost uniform field inside, very similar to a bar magnet outside. A solenoid is the electrical equivalent of a bar magnet. The right-hand grip rule applied to the entire winding gives: curl fingers in the direction of current flow around the windings, thumb points along the field direction inside (and out of the "N pole" end).

The field of a long solenoid is (approximately) $B = \mu_0 n I$B=μ0​nI, where $n$n is the number of turns per unit length. This formula is beyond the OCR H556 spec, but it's worth knowing the qualitative content: more turns per metre, more current, stronger field.

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Two Parallel Wires

Two long parallel wires carrying currents $I_1$I1​ and $I_2$I2​ separated by a distance $d$d exert forces on each other:

• Parallel currents attract.
• Antiparallel currents repel.

Reason: wire 1 produces a field at the location of wire 2; wire 2's current then experiences a force $F = BI_2 L$F=BI2​L in that field. By the right-hand grip rule, parallel currents end up with $\vec{F}$F pulling them together; antiparallel currents end up with $\vec{F}$F pushing them apart.

This was the basis of the historical (pre-2019) definition of the ampere: "one ampere is the current in two very long parallel wires $1$1 m apart that exerts a force of $2 \times 10^{-7}$2×10−7 N per metre on each other". OCR does not require the quantitative formula ($F/L = \mu_0 I_1 I_2/(2\pi d)$F/L=μ0​I1​I2​/(2πd)), but does expect you to know the direction rules and that the force can be predicted by Fleming's left-hand rule applied twice (field of one wire, current of the other).

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