Magnetic Fields Produced by Current-Carrying Wires and Solenoids (HSC SSCE Physics): Revision Notes

Magnetic Fields Produced by Current-Carrying Wires and Solenoids

Introduction

In 1819, Hans Christian Øersted made a groundbreaking discovery: a current-carrying wire creates a magnetic field. This observation revolutionised our understanding of electromagnetism and led to the development of motors, generators, and many other useful devices we use today.

While permanent magnets made from ferromagnetic materials are convenient for producing small magnetic fields, when a large field is needed, we use an electromagnet. An electromagnet consists of a current-carrying coil of wire and can be switched on and off by controlling the current.

Different arrangements of current-carrying wires can create magnetic fields of different strengths and shapes. In this note, we'll explore two important configurations: straight wires and solenoids (coils).

Magnetic field due to a current-carrying wire

Key observations by Biot and Savart

Jean-Baptiste Biot and Félix Savart performed extensive experiments with magnets and current-carrying wires. They discovered three important relationships for the magnetic field at a point $P$ some distance from a long straight current-carrying wire:

1. Direction: The magnetic field is perpendicular to both the direction of the current and to a line between the wire and point $P$.
2. Distance relationship: The magnitude of the field is inversely proportional to the distance from the wire to $P$. This means as you move further from the wire, the field becomes weaker.
3. Current relationship: The magnitude of the field is directly proportional to the current. Doubling the current doubles the field strength.

Mathematical formula

These observations can be expressed mathematically as:

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

where:

• $B$ is the magnitude of the magnetic field (in tesla, T)
• $I$ is the current in the wire (in amperes, A)
• $r$ is the perpendicular distance from the wire (in metres, m)
• $\mu_0$ is the permeability of free space

The permeability of free space

The constant $\mu_0$ is called the permeability of free space and has the value:

$\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$

This constant serves the same function for magnetic fields as the permittivity of free space ($\varepsilon_0$) does for electric fields. It determines the strength of a field created by a given current in vacuum and ensures the units work out correctly.

Units of magnetic field

Magnetic field is measured in tesla (symbol: T), named after Nikola Tesla who made important contributions to electricity and magnetism. The tesla can be expressed in fundamental SI units as:

$1 \text{ T} = 1 \text{ kg s}^{-2} \text{ A}^{-1}$

Worked example: Calculating field strength

Direction of the magnetic field

The magnetic field around a current-carrying wire forms circular loops. To determine the direction of these field lines, we use the right-hand rule (also called the "rule of thumb"):

Right-hand rule procedure:

1. Point your right thumb in the direction of the current
2. Your fingers naturally curl in the direction of the magnetic field lines

When the current is coming towards you (out of the page), the magnetic field lines circle anticlockwise. When the current is going away from you (into the page), the field lines circle clockwise.

Drawing magnetic field lines

When drawing field line diagrams for a current-carrying wire:

• Draw circular loops centred on the wire
• Space the loops further apart as you move away from the wire (since field strength decreases with distance)
• Add arrows to show the direction of the field
• Draw a sensible number of field lines (not too many, not too few)

The density (closeness) of field lines represents field strength—closer lines indicate stronger field.

Magnetic field due to a current-carrying solenoid

What is a solenoid?

A solenoid is a coil of wire wound into a cylindrical shape. It's one of the most effective ways to create a large, uniform magnetic field. By coiling a single wire into many loops, we essentially combine the magnetic fields from each loop to produce a strong total field.

How solenoids create magnetic fields

Each loop of wire in the solenoid creates its own circular magnetic field (just like a single wire). Inside the coil, these individual fields add together to produce:

• A large, uniform magnetic field with straight, parallel field lines
• A field strength that increases with the number of turns (loops)
• A very weak field outside the coil

In a tightly wound solenoid (loops close together), the internal field becomes very uniform, with field lines running straight and parallel along the axis of the coil.

Comparison to a bar magnet

The magnetic field pattern of a solenoid is remarkably similar to that of a bar magnet. Just as a bar magnet has north and south poles at its ends, a solenoid also has:

• A north pole at one end
• A south pole at the other end

The field lines emerge from the north pole and curve around to enter the south pole, forming closed loops—just like with a permanent magnet.

Mathematical formula for solenoid field

The magnetic field strength inside a solenoid depends on three factors:

$B = \frac{\mu_0 N I}{L}$

where:

• $B$ is the magnetic field strength inside the solenoid (in tesla, T)
• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7}$ T m A$^{-1}$)
• $N$ is the number of turns (loops) of wire
• $I$ is the current through the wire (in amperes, A)
• $L$ is the length of the solenoid (in metres, m)

Worked example: Calculating current in a solenoid

Investigation 14.5: Investigating magnetic fields due to currents

This practical investigation allows you to explore the magnetic fields created by different current configurations.

Aim

To investigate the magnetic fields produced by current-carrying wires in different arrangements (straight wire, coil/solenoid, and parallel wires).

Materials

• Magnetic field meter or compass (alternatively, a smartphone with compass and magnetic field meter app)
• DC power supply
• Cardboard tube
• Light globe of same voltage as power supply (e.g. 12 V globe for 12 V supply)
• Insulated wire (at least 2 m long)
• Connectors (such as alligator clips)
• Tape
• At least 1 m of clear bench space

Risk assessment

Be careful not to set the voltages and currents too high, as there is a danger of electric shock.

Method

1. Preparation: If using a magnetic field meter or app, read the instructions and ensure you know how to use it correctly, including whether it needs to be aligned with the field.
2. Baseline measurement: Before connecting any equipment, measure the background magnetic field strength (Earth's magnetic field) or note the compass needle direction. Record this result.
3. Circuit setup: Connect the wire to the light globe at one end and the power supply at the other. Connect the other terminal of the globe to the power supply to complete the circuit.

1. Straight wire investigation: Stretch out the wire so you have a straight section well away from the power supply. You may tape a section to the bench to keep it straight.
2. Measurement: Turn on the power supply. The light should glow, confirming current flow. Measure the field around the straight wire or note compass deflections. Record how the field varies above and to the sides of the wire. Note the current direction.
3. Reverse current: Observe what happens when you reverse the direction of current through the wire.
4. Solenoid investigation: Turn off the power and disconnect the wire from the power supply (leave it connected to the light globe). Wrap the wire around the cardboard tube to form a coil, always wrapping in the same direction. Secure with tape. Leave enough wire free to reconnect to the power supply.
5. Solenoid measurements: Connect the coiled wire to the power supply. Repeat measurements inside the tube, at its ends, and to the side.
6. Parallel wires investigation: Lay the wire back and forth on the bench in parallel lines, taping down as needed.
7. Additional investigations: Other wire arrangements can also be explored.

Results

You should have records of the magnetic field for:

• A long straight current-carrying wire
• A solenoid (coil)
• Parallel current-carrying wires

For each configuration, note how the field strength varies with position and how it changes when current direction is reversed.

Expected observations

• Straight wire: Circular field lines around the wire, field strength decreasing with distance
• Solenoid: Strong, uniform field inside the coil; much weaker field outside; field pattern similar to a bar magnet
• Field direction: Reversing current reverses the direction of the magnetic field

Remember!