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Magnetic Fields due to Currents – Cambridge A‑Level Physics 9702

Magnetic Fields due to Currents

Learning Objective

Students will be able to sketch and describe the magnetic‑field patterns produced by:

a long straight current‑carrying wire,

a flat circular coil, and

a long solenoid.

1. Long Straight Wire

A current \$I\$ flowing in an infinitely long straight conductor creates concentric circular magnetic field lines around the wire.

Using the right‑hand rule: point the thumb in the direction of conventional current; the curled fingers show the direction of the magnetic field \$\\mathbf{B}\$.

The magnitude of the field at a distance \$r\$ from the centre of the wire is given by Ampère’s law:

\$B = \frac{\mu_0 I}{2\pi r}\$

where \$\\mu_0 = 4\pi\\times10^{-7}\\,\\text{T·m·A}^{-1}\$.

Suggested diagram: concentric circles around a straight wire with arrows indicating the direction of \$\\mathbf{B}\$ according to the right‑hand rule.

2. Flat Circular Coil

A single loop of wire carrying current \$I\$ produces a magnetic field that is strongest at the centre of the loop and weakens with distance along the axis.

On the axis of the coil (distance \$x\$ from the centre), the field magnitude is:

\$B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}\$

where \$R\$ is the radius of the coil.

Key features of the field pattern:

Field lines emerge from one face of the coil and re‑enter the opposite face, forming closed loops.

Inside the coil the field is approximately uniform and directed perpendicular to the plane of the coil.

Outside the coil the field spreads out and becomes weaker with distance.

Suggested diagram: a circular loop with field lines emerging from the centre, passing through the loop, and returning on the opposite side; arrows indicate direction using the right‑hand rule.

3. Long Solenoid

A solenoid is a coil of many turns of wire. When a current \$I\$ flows, the magnetic field inside is nearly uniform and parallel to the axis, while the external field is weak.

For a solenoid of \$n\$ turns per unit length, the internal field magnitude is:

\$B = \mu_0 n I\$

If the solenoid has \$N\$ total turns and length \$L\$, then \$n = \frac{N}{L}\$.

Characteristics of the field pattern:

Inside: parallel field lines, almost uniform, directed according to the right‑hand rule (thumb points along the solenoid axis in the direction of current).

Ends: field lines bulge outward, forming a pattern similar to that of a bar magnet (north and south poles).

Outside: field is much weaker and spreads out, resembling the field of a dipole.

Suggested diagram: a cylindrical solenoid with dense, parallel field lines inside and curved lines exiting the ends, labelled north and south poles.

Comparison of the Three Configurations

Feature	Long Straight Wire	Flat Circular Coil	Long Solenoid
Field Geometry	Concentric circles around the wire	Loops emerging from one face and entering the other	Uniform, parallel lines inside; weak, dipole‑like outside
Direction Rule	Right‑hand thumb = current	Right‑hand thumb = current through loop	Right‑hand thumb = current direction along coil axis
Field Strength Formula	\$B = \dfrac{\mu_0 I}{2\pi r}\$	\$B = \dfrac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}\$ (on axis)	\$B = \mu_0 n I\$ (inside)
Typical Applications	Transmission lines, magnetic sensors	Electromagnets, inductors	Electromagnets, transformers, MRI

Key Points to Remember

Use the right‑hand rule consistently for direction of \$\\mathbf{B}\$.

For a straight wire, the field falls off as \$1/r\$.

For a single loop, the field is strongest at the centre and decreases with axial distance.

For a long solenoid, the field inside is essentially uniform and independent of the radius of the coil.

All magnetic field lines form closed loops; they never begin or end in space.

sketch magnetic field patterns due to the currents in a long straight wire, a flat circular coil and a long solenoid