explain the origin of the forces between current-carrying conductors and determine the direction of the forces

1. Introduction

When an electric current flows through a conductor it produces a magnetic field in the surrounding space.

The magnetic field can exert forces on other currents or on moving charges. These ideas form the core of Cambridge International AS & A Level Physics (Topic 20 – Magnetic fields) and are the basis for later topics such as electromagnetic induction, motors, and the definition of the ampere.

2. The magnetic field

2.1 Definition and basic properties

• Magnetic flux density \( \mathbf{B} \) is a vector field that describes the magnetic influence of currents and magnetic materials.
• Direction: the way a north‑pole of a tiny test magnet would point.
• Field‑line conventions:

  • Lines are tangent to the direction of \( \mathbf{B} \) at every point.
  • The density of lines is proportional to the field strength.
  • Lines form closed loops; they never start or end.

• SI unit: the tesla (T), where \(1\;\text{T}=1\;\text{N·A}^{-1}\text{m}^{-1}=1\;\text{kg·s}^{-2}\text{A}^{-1}\).

2.2 \( \mathbf{B} \) vs \( \mathbf{H} \) and magnetic materials

The two are related by the magnetic permeability \( \mu \):

\[

\mathbf{B}= \mu \mathbf{H},\qquad \mu = \mu{0}\mu{r},

\]

where \( \mu{0}=4\pi\times10^{-7}\;\text{T·m·A}^{-1} \) is the permeability of free space and \( \mu{r} \) is the relative permeability of the material.

Effect of magnetic materials

• Ferromagnetic (e.g. iron): \( \mu_{r}\gg1 \) → field lines are concentrated inside the material.
• Paramagnetic: \( \mu_{r}>1 \) (weak attraction).
• Diamagnetic: \( \mu_{r}<1 \) (weak repulsion).

2.3 Magnetic fields produced by currents

2.3.1 Straight, infinitely long conductor

From the Biot–Savart law, the magnitude of the field at a perpendicular distance \( r \) from a long straight wire carrying current \( I \) is

\[

B = \frac{\mu_{0} I}{2\pi r}.

\]

Direction: right‑hand grip rule – thumb points in the direction of conventional current, fingers curl in the direction of \( \mathbf{B} \).

2.3.2 Circular loop of radius \( R \)

For a single loop carrying current \( I \), the field on the axis a distance \( x \) from the centre is

\[

B{\text{axis}} = \frac{\mu{0} I R^{2}}{2\,(R^{2}+x^{2})^{3/2}}.

\]

At the centre (\( x=0 \)) this reduces to

\[

B{\text{centre}} = \frac{\mu{0} I}{2R}.

\]

Direction: curl the fingers in the sense of the current; the thumb points along the field inside the loop.

2.3.3 Solenoid ( \( n \) turns per unit length )

Inside a long solenoid the field is almost uniform and parallel to the axis:

\[

B{\text{inside}} = \mu{0} n I.

\]

Outside an ideal (infinitely long) solenoid the field is essentially zero. The direction follows the same right‑hand rule as for a single loop.

3. Forces involving magnetic fields

3.1 Lorentz force on a moving charge

\[

\mathbf{F}=q\,\mathbf{v}\times\mathbf{B}.

\]

• Magnitude: \( F = qvB\sin\theta \) ( \( \theta \) is the angle between \( \mathbf{v} \) and \( \mathbf{B} \) ).
• Direction: for a positive charge, use the right‑hand rule (point fingers along \( \mathbf{v} \), rotate toward \( \mathbf{B} \); thumb gives \( \mathbf{F} \)). For an electron the force is opposite.

3.2 Deriving the force on a current‑carrying conductor

Consider a small segment of wire of length \( \mathrm{d}\mathbf{L} \) containing charge carriers of density \( n \) (charges per unit volume) moving with drift speed \( \mathbf{v}_{d} \). The number of carriers in the segment is \( nA\,\mathrm{d}L \) (where \( A \) is the cross‑sectional area) and the total charge is \( q = n e A\,\mathrm{d}L \).

Each carrier experiences the Lorentz force \( \mathrm{d}\mathbf{F}=q\,\mathbf{v}_{d}\times\mathbf{B} \). Summing over the segment gives

\[

\mathrm{d}\mathbf{F}= (n e A\,\mathrm{d}L)\,\mathbf{v}_{d}\times\mathbf{B}.

\]

But the current is \( I = n e A v_{d} \). Substituting,

\[

\mathrm{d}\mathbf{F}= I\,\mathrm{d}\mathbf{L}\times\mathbf{B}.

\]

Integrating over the whole conductor of length \( \mathbf{L} \) yields the familiar expression

\[

\boxed{\mathbf{F}= I\,\mathbf{L}\times\mathbf{B}}.

\]

Hence the magnitude is \( F = I L B \sin\theta \) and the direction follows the right‑hand rule (thumb → \( I\mathbf{L} \), fingers → \( \mathbf{B} \), palm → \( \mathbf{F} \)).

3.3 Interaction between two parallel current‑carrying conductors

3.3.1 Origin of the force

1. Conductor 1 (current \( I_{1} \)) creates a magnetic field at the location of conductor 2:

   \[

   B{1}= \frac{\mu{0} I_{1}}{2\pi d},

   \]

   where \( d \) is the centre‑to‑centre separation.

2. Conductor 2 (current \( I_{2} \)) experiences a force

   \[

   F{21}= I{2} L B{1}= \frac{\mu{0} I{1} I{2} L}{2\pi d},

   \]

   directed by the right‑hand rule for \( \mathbf{L}{2}\times\mathbf{B}{1} \).

3. By Newton’s third law, conductor 1 feels an equal and opposite force.

3.3.2 Direction of the force

3.3.3 Numerical example

Two long wires are 4 cm apart and each carries 5 A in the same direction. Find the force per metre of length.

\[

\frac{F}{L}= \frac{\mu{0} I{1} I_{2}}{2\pi d}

= \frac{4\pi\times10^{-7}\times5\times5}{2\pi\times0.04}

= 1.25\times10^{-4}\;\text{N m}^{-1}.

\]

Thus the wires attract each other with a force of \(0.125\;\text{mN}\) per metre of length.

4. Definition of the ampere

One ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length and negligible circular cross‑section, placed 1 m apart in vacuum, would produce a force of \(2\times10^{-7}\;\text{N}\) per metre of length on each conductor.

This definition follows directly from the formula \(F/L = \mu{0}I^{2}/(2\pi d)\) with \(I{1}=I_{2}=1\;\text{A}\) and \(d=1\;\text{m}\).

5. Applications and links to later topics

• Electromagnetic induction: A changing current alters the magnetic field, inducing an emf in a nearby circuit (Faraday’s law).
• Electric motors and generators: The force \( \mathbf{F}=I\mathbf{L}\times\mathbf{B} \) on a current‑carrying coil in a magnetic field produces torque.
• Inductance: The self‑field of a current‑carrying coil opposes changes in current, leading to the concept of inductance \( L \).
• Magnetic materials: Knowledge of \( \mu_{r} \) is essential for designing electromagnets, transformers and magnetic shielding.

6. Summary of key points

• \( \mathbf{B} \) is the magnetic flux density (unit T); \( \mathbf{H} \) is the magnetic field intensity (unit A·m\(^{-1}\)). They are related by \( \mathbf{B}= \mu\mathbf{H} \).
• Current‑carrying conductors produce circular magnetic fields; direction is given by the right‑hand grip rule.
• Magnetic‑field formulas:

  • Straight wire: \( B=\dfrac{\mu_{0}I}{2\pi r} \)
  • Circular loop (centre): \( B=\dfrac{\mu_{0}I}{2R} \)
  • Solenoid (inside): \( B=\mu_{0}nI \)

• Lorentz force on a charge: \( \mathbf{F}=q\mathbf{v}\times\mathbf{B} \).
• Force on a current element (derived from Lorentz): \( \mathbf{F}=I\mathbf{L}\times\mathbf{B} \) → \( F=ILB\sin\theta \).
• Two parallel conductors:

  • Force per metre: \( \displaystyle\frac{F}{L}= \frac{\mu{0}I{1}I_{2}}{2\pi d} \)
  • Same‑direction currents → attraction; opposite currents → repulsion.

• The ampere is defined via the force between parallel conductors, linking magnetic concepts to the SI system.