Concept of a magnetic field - Physics (9702) - Cambridge A-Level

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Cambridge A Level Physics (9702): Concept of a Magnetic Field

Hello future Physicists! Welcome to the exciting world of Magnetism. This chapter builds directly on your knowledge of forces and electricity, showing how moving charges (currents) create invisible fields that exert powerful forces. Don't worry if this seems tricky at first; we will break down the fundamental concepts step-by-step, using simple analogies to make the invisible visible!

Understanding magnetic fields is crucial because they underpin almost all modern technology—from electric motors and loudspeakers to mass spectrometers and power generation. Let's dive in!

20.1 The Concept of a Magnetic Field

What is a Magnetic Field?

Just as a mass creates a gravitational field, and a charge creates an electric field, a magnetic field is defined as a region in space where a magnetic force can be experienced.

A magnetic field is an example of a field of force.
It is produced either by moving electric charges (electric currents) or by permanent magnets.

Representing Magnetic Fields: Field Lines

We represent magnetic fields using magnetic field lines (also called magnetic flux lines). These lines show two things:

Direction: The field lines always point away from the North pole (N) and towards the South pole (S) of a magnet, or in the direction a North pole would move.
Strength: The magnetic field is stronger where the lines are drawn closer together (denser).

Quick Review of Field Line Conventions:

Lines never cross.
The field inside a solenoid (a long coil) is strong, uniform, and parallel.

Analogy: Think of magnetic field lines like contour lines on a map. Where the lines are close, the force (slope) is steep; where they are spread out, the force is weak.

Key Takeaway (20.1): Magnetic fields are force fields created by magnets or moving charges. We map them using field lines that indicate direction (N to S) and strength (density).

20.2 Force on a Current-Carrying Conductor

When a wire carrying a current (i.e., moving charges) is placed inside an external magnetic field, it experiences a force. This is the basis of how electric motors work!

The Magnetic Force Formula

The magnitude of the force $F$ acting on a straight conductor of length $L$ carrying current $I$ in a uniform magnetic field of flux density $B$ is given by:

$$ F = BIL \sin \theta $$

Where:

$F$ is the Force (in Newtons, N).
$B$ is the Magnetic Flux Density (in Tesla, T).
$I$ is the Current (in Amperes, A).
$L$ is the Length of the conductor immersed in the field (in meters, m).
$\theta$ is the angle between the direction of the current $I$ and the magnetic field $B$.

Crucial Point: The force is maximum when the wire is perpendicular to the field ($\theta = 90^\circ$, so $\sin 90^\circ = 1$). The force is zero when the wire is parallel to the field ($\theta = 0^\circ$, so $\sin 0^\circ = 0$).

Defining Magnetic Flux Density (B)

The magnetic flux density, B, is a measure of the strength of the magnetic field.

Definition: Magnetic Flux Density (B) is defined as the force acting per unit current per unit length on a wire placed at right-angles ($\theta = 90^\circ$) to the magnetic field.

Using the formula $F = BIL \sin \theta$, if we set $I = 1\text{ A}$, $L = 1\text{ m}$, and $\theta = 90^\circ$, then $B = F$.

The SI unit for magnetic flux density is the Tesla (T).

Defining the Tesla (T)

One Tesla is the magnetic flux density that produces a force of 1 Newton on a wire of 1 metre length carrying a current of 1 Ampere, placed perpendicular to the field.

Determining Direction: Fleming's Left-Hand Rule (LHR)

Since Force, Field, and Current are all vector quantities, we need a rule to find their relative directions. We use Fleming's Left-Hand Rule (LHR):

Stretch your LEFT hand so your thumb, forefinger, and middle finger are mutually perpendicular (at 90° to each other).
Forefinger (First Finger): Points in the direction of the Field (B).
Middle Finger: Points in the direction of the Current (I).
Thumb: Points in the direction of the Force (F) or motion.

🧠 Memory Aid: FBI

F: Thumb = Force
B: Forefinger = B Field
I: Middle finger = I Current (Conventional current, positive flow)

Key Takeaway (20.2): The force on a conductor is $F = BIL \sin \theta$. The strength B is measured in Tesla (T). Use Fleming's Left-Hand Rule to determine the direction of the force (FBI).

20.3 Force on a Moving Charge

A current is simply a flow of charges. If a charge $Q$ moves through a magnetic field $B$ with velocity $v$, it experiences a force.

The Magnetic Force Formula for a Single Charge

The force $F$ acting on a single charge $Q$ moving at velocity $v$ in a magnetic field $B$ is:

$$ F = BQv \sin \theta $$

Where $\theta$ is the angle between the velocity vector $v$ and the magnetic field vector $B$.

Determining Direction for Charges

We still use Fleming's Left-Hand Rule (LHR), but we must be careful:

If the particle is positive (e.g., proton, $\text{positive ion}$), the direction of $v$ is the direction of the current $I$. Use LHR normally.
If the particle is negative (e.g., electron), the direction of the current $I$ is opposite to the direction of $v$.

Motion of a Charged Particle (Perpendicular Field)

If a charged particle enters a uniform magnetic field at a right angle ($\theta = 90^\circ$), the magnetic force $F = BQv$ is always perpendicular to the velocity $v$.

A force always perpendicular to motion causes the particle to move in a circle.
The magnetic force provides the necessary centripetal force.

$$ F_{\text{magnetic}} = F_{\text{centripetal}} $$ $$ BQv = \frac{mv^2}{r} $$

This equation allows us to find the radius $r$ of the circular path: $$ r = \frac{mv}{BQ} $$

Did you know? This principle is used in devices called mass spectrometers, which separate ions based on their mass-to-charge ratio ($m/Q$).

The Hall Effect and Velocity Selection

The Hall Effect (AS Level extension)

When a current-carrying conductor (usually a semiconductor slice) is placed perpendicular to a magnetic field, the magnetic force pushes the moving charge carriers (electrons or holes) to one side. This separation of charges creates a voltage across the sides of the conductor, known as the Hall Voltage, $V_H$.

Importance: Measuring $V_H$ allows us to determine the sign and density of the charge carriers in the material.

The expression for the Hall voltage is:

$$ V_H = \frac{BI}{ntq} $$

Where:

$B$ is the magnetic flux density.
$I$ is the current through the slice.
$t$ is the thickness of the slice (in the direction of the B field).
$n$ is the number density of charge carriers (number of carriers per unit volume).
$q$ is the charge of the carrier.

A device based on this principle, called a Hall probe, is used to measure magnetic flux density $B$.

Velocity Selection (Combining E and B Fields)

A velocity selector uses crossed uniform electric fields ($E$) and magnetic fields ($B$) perpendicular to each other and perpendicular to the particle's direction of motion.

Electric Force: $F_E = QE$
Magnetic Force: $F_B = BQv$

If the forces are balanced ($F_E = F_B$) and opposite in direction, the charged particle passes through the fields undeflected.

$$ QE = BQv $$

The charge $Q$ cancels out, meaning only particles moving at a specific velocity $v$ will pass through:

$$ v = \frac{E}{B} $$

Key Takeaway (20.3): A force $F = BQv \sin \theta$ acts on a moving charge. Perpendicular fields cause circular motion. The Hall effect (based on $V_H = BI/(ntq)$) helps measure field strength, and crossed E and B fields can select particles with specific velocities ($v=E/B$).

20.4 Magnetic Fields Due to Currents

We established that moving charges create magnetic fields. Now let's look at the patterns these fields form around common current structures.

Rule for Direction: Right-Hand Grip Rule (RHGR)

To determine the direction of the magnetic field lines created by a current (I):

Grasp the conductor (wire, coil, or solenoid) with your RIGHT hand.
Point your Thumb in the direction of the Conventional Current (I).
Your curled Fingers indicate the direction of the Magnetic Field Lines (B).

Field Patterns to Sketch

1. Long Straight Wire

The magnetic field lines are concentric circles centred on the wire.
The strength decreases as you move further away from the wire.

2. Flat Circular Coil

Near the wire, the field lines are circular (following the RHGR).
Through the centre of the coil, the field lines are straight and parallel (and perpendicular to the plane of the coil).
The overall pattern resembles that of a small permanent bar magnet.

3. Long Solenoid (Coil of many turns)

Inside a long solenoid, the magnetic field is uniform and parallel to the axis of the solenoid.
Outside, the field pattern is similar to a bar magnet (with one end acting as North and the other as South).

Effect of a Ferrous Core

If a material like iron (a ferrous core) is placed inside a solenoid, the magnetic field strength is significantly increased. This is because the magnetic properties of the iron align strongly with the field, leading to a much higher flux density (B). This creates a powerful electromagnet.

Forces Between Current-Carrying Conductors

Since Current A creates a magnetic field, and Current B experiences a force in that field, two parallel current-carrying wires will exert a force on each other.

Step-by-step determination of force direction:

Use the Right-Hand Grip Rule to find the direction of the field ($B_1$) created by Wire 1 at the location of Wire 2.
Use Fleming's Left-Hand Rule to find the force ($F_2$) on Wire 2 due to the field $B_1$.

Rule Summary:

If the currents flow in the SAME DIRECTION, the wires will attract each other.
If the currents flow in OPPOSITE DIRECTIONS, the wires will repel each other.

Common Mistake Alert: Always be clear on which rule you are using! RHGR determines the FIELD produced by the current. LHR determines the FORCE exerted by an external field on the current.

Key Takeaway (20.4): Currents generate magnetic fields whose direction is found by the RHGR. Parallel currents attract if flowing together and repel if flowing apart. A ferrous core greatly boosts the magnetic field strength inside a solenoid.