# 14.1.1 Wires, Coils and Solenoids

In 1820, Hans Christian Rested discovered, by accident, that an electric current could influence compass needles. Thus began the study of electromagnetism.

Well, it’s not science unless it is quantified. So the magnetic field is quantified by what’s called the magnetic flux density B. It is a vector quantity, with the SI unit tesla (symbol T) [1]. Tabulated below are the typical magnitude of magnetic flux density produced by different types of magnets.

Long Straight Wire

An electric current running in a long straight wire creates a concentric circular magnetic field outside the wire. Its direction (clockwise or counter-clockwise) depends on the direction of the current in the wire.

To use the right hand grip rule (RHGR), point your thumb in the direction of the (conventional) current I and curl your fingers. The direction of your curled fingers will mirror the circumferential direction of the magnetic field B.

The magnitude of the magnetic flux density produced by a long straight wire at distance r from the wire is given by the formula

$\displaystyle B=\frac{{{{\mu }_{o}}I}}{{2\pi r}}$

Notes:

• The constant $\displaystyle {{\mu }_{0}}=4\pi \times {{10}^{{-7}}}\text{ H }{{\text{m}}^{{-1}}}$ is called the permeability of free space. Roughly speaking, permeability is a measure of how magnetizable a material is. Iron (99.8% pure) for example is 5000 times more permeable compared to vacuum.
• B decreases with distance from r the wire. As such, the field pattern is drawn with the circles more and more spaced out as we go further and further away from the wire.

Flat Circular Coil

If we bend the wire into a flat circular coil, the magnetic field produced by each section of the coil will superpose inside the loop to produce a stronger magnetic field. The magnetic flux density at the centre of the coil is given by the formula

$\displaystyle B=\frac{{{{\mu }_{0}}I}}{{2r}}$

where r is the radius of the coil and I the current in the coil.

To use the right hand grip rule (RHGR), curl your fingers in the direction of the (conventional) current I in the coil and stick out your thumb. The direction of your thumb is in the direction of the magnetic field B inside the coil.

We can stack N number of identical coils one on top of another to obtain an even stronger field. As long as the coil remains flat enough (negligible thickness compared to the radius r of the coil), the magnetic flux density at the centre of the coils is given by the formula

$\displaystyle B=\frac{{{{\mu }_{0}}NI}}{{2r}}$

Long Solenoid

The easiest way to obtain a uniform magnetic field is toe keep adding more turns to a flat circular coil until a solenoid is formed.

If the turns are wound closely enough, and the solenoid long enough, the resultant field inside the solenoid is actually uniform[2].

To use the right hand grip rule (RHGR), curl your fingers in the direction of the (conventional) current I and you thumb will be pointing the direction of the magnetic field B inside the solenoid.

The magnitude of the magnetic flux density inside the solenoid is given by the formula

$\displaystyle B={{\mu }_{o}}nI$

Notes:

• n is number of turns per unit length. e.g. 20000 turns per metre.
• The field pattern of a solenoid resembles that of a bar magnet. The field lines go from the North pole to the South pole outside the solenoid, but South to North inside the solenoid.
• The density of the field lines is an indication of the strength of the magnetic field. From the field pattern, you can tell that the field is strongest (and uniform) inside the solenoid. Outside the solenoid, the strongest field is found at the poles.[3]

Demonstration

Magnetic Field Patterns

Concept Test

2801

2804

2803

Interesting

Magnetic Toys

[1] Magnetic field is also quantified by another quantity called magnetic field strength H. In vacuum, H and B related through the magnetic permeability $\displaystyle H=\frac{B}{{{{\mu }_{o}}}}$. In magnetised material the relation is more complicated. H is not in the H2 syllabus.

[2] the electric counterpart would be the uniform electric field produced by large parallel plates

[3] In fact, you can use simple logic to deduce that the strength right at edge of the long solenoid is exactly half the strength inside the long solenoid.