Physics (9702) | Magnetic fields

Welcome to Chapter 20: Magnetic Fields!

Hello future physicists! This chapter dives into one of the most fundamental forces in the Universe: magnetism. We will learn how moving charges (currents) create magnetic fields, how these fields exert forces, and how we can use this interaction to generate electricity—a process called electromagnetic induction.

Don't worry if magnetism feels a bit abstract compared to forces you can see. We’ll use clear rules (like the famous Fleming's rules!) to make the directions and calculations straightforward. By the end of this chapter, you’ll understand the physics behind everything from electric motors and particle accelerators to credit card readers!

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20.1 Concept of a Magnetic Field

What is a Magnetic Field?

A magnetic field (represented by the symbol $B$) is a region of space where a magnetic force can be experienced. It is an example of a field of force, just like a gravitational field or an electric field.

Magnetic fields are produced in two ways:

• By permanent magnets (like a fridge magnet).
• By moving charges (electric currents).

Representing Magnetic Fields: Field Lines

We represent magnetic fields using field lines (sometimes called flux lines).

• The lines show the direction a free North pole would move.
• Field lines travel from the North pole (N) to the South pole (S) outside the magnet.
• Where the lines are closer together, the magnetic field is stronger.
• Field lines never cross.

When sketching field diagrams, you must be able to represent direction in 3D:
• A dot (\(\cdot\)) represents the field direction coming out of the page.
• A cross (\(\times\)) represents the field direction going into the page.

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20.2 Force on a Current-Carrying Conductor

The Motor Effect

If you place a current-carrying wire in an external magnetic field, the wire experiences a force. This is often called the motor effect, as this is the fundamental principle that makes electric motors work!

Calculating the Force (\(F\))

The magnitude of the force ($F$) acting on a wire depends on four factors:

\[F = BIL \sin \theta\]

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 wire exposed to the field (in metres, m).
• \(\sin \theta\) is the angle between the wire and the magnetic field direction.

Important Note: 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.

If \(\theta = 90^\circ\), then \(F = BIL\). Rearranging gives:

\[B = \frac{F}{IL}\]

The SI unit for magnetic flux density is the Tesla (T).
1 Tesla is the magnetic flux density that produces a force of 1 N on 1 m of wire carrying a current of 1 A, placed perpendicular to the field.

Determining the Direction: Fleming's Left-Hand Rule

The force, field, and current are all mutually perpendicular when the force is maximum. We use Fleming's Left-Hand Rule to find the direction of the force ($F$):

1. Use your LEFT hand (for the Left-Hand Rule, Motor Effect).
2. Point your Thumb in the direction of the Force ($F$).
3. Point your First Finger in the direction of the Field ($B$).
4. Point your Second Finger in the direction of the conventional Current ($I$).

Memory Aid (FBI):
• First finger = B field
• Index finger = I current (Charge carriers are positive)
• Thumb = Force

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20.3 Force on a Moving Charge

Since current ($I$) is just the movement of charge ($Q$), it stands to reason that an individual charged particle moving through a magnetic field will also experience a force.

Calculating the Force on a Charge

The force ($F$) acting on a moving charge ($Q$) with velocity ($v$) is:

\[F = BQv \sin \theta\]

Where:
• \(F\) is the force (N).
• \(B\) is the Magnetic Flux Density (T).
• \(Q\) is the magnitude of the charge (C).
• \(v\) is the velocity of the charge (m s\(^{-1}\)).
• \(\sin \theta\) is the angle between the velocity vector and the magnetic field direction.

Note: For a positive charge, the direction of $v$ is used for $I$ in the Left-Hand Rule. If the particle is an electron (negative charge), the direction of the force will be opposite to that predicted by the Left-Hand Rule, or you can use the direction of the electron's motion as the direction of $I$ and apply the Left-Hand Rule normally.

Motion of a Charged Particle in a Uniform Field

Consider a charged particle entering a uniform magnetic field ($B$) perpendicular to its velocity (\(\theta = 90^\circ\)).

1. The force $F = BQv$ acts perpendicular to the velocity $v$.
2. This perpendicular force acts as a centripetal force, causing the particle to move in a circle at a constant speed.
3. We can equate the magnetic force to the centripetal force:

\[BQv = \frac{mv^2}{r}\]

4. This allows us to find the radius ($r$) of the circular path:

\[r = \frac{mv}{BQ}\]

Did you know? This principle is used in particle accelerators (like cyclotrons) and mass spectrometers to identify particles based on their mass-to-charge ratio.

Velocity Selection (The E and B Field Combined)

If a particle passes through a region containing both a uniform electric field ($E$) and a uniform magnetic field ($B$) that are perpendicular to each other and perpendicular to the particle's velocity, the fields can be used as a velocity selector.

1. The Electric Force ($F_E$) is \(F_E = QE\).
2. The Magnetic Force ($F_B$) is \(F_B = BQv\).

If the fields are arranged such that these two forces oppose each other, a particle will only pass straight through (undeflected) if the forces are perfectly balanced:

\[F_E = F_B\] \[QE = BQv\]

The charge $Q$ cancels out, meaning the required speed ($v$) is:

\[v = \frac{E}{B}\]

Only particles travelling at this specific speed $v$ will pass through without being deflected.

The Hall Effect

The Hall Effect demonstrates that the current in a conductor is indeed due to moving charges.

Imagine a flat strip of conductor carrying current $I$ placed perpendicular to a magnetic field $B$:

1. The moving charge carriers (electrons or holes) experience a magnetic force, \(F = BQv\), which pushes them sideways.
2. These charges pile up on one side of the conductor, creating a charge separation.
3. This charge separation generates a potential difference (voltage) across the sides of the strip, known as the Hall Voltage ($V_H$).
4. This Hall Voltage creates an electric field ($E$) that opposes the magnetic force.
5. Equilibrium is reached when the electric force balances the magnetic force ($F_E = F_B$), and charges flow straight again.

The magnitude of the Hall voltage is given by the expression:

\[V_H = \frac{BI}{ntq}\]

Where:
• $B$ is the magnetic flux density (T).
• $I$ is the current (A).
• $n$ is the number density of charge carriers (carriers per volume, m\(^{-3}\)).
• $t$ is the thickness of the conductor strip (m), measured parallel to the magnetic field.
• $q$ is the charge of the carrier (C).

Application: A device called a Hall probe uses the Hall effect to measure magnetic flux density ($B$) directly and accurately. Since $V_H$ is directly proportional to $B$, measuring the voltage gives the field strength.

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20.4 Magnetic Fields Due to Currents

Moving charges create their own magnetic fields. We need to know the shape and direction of the fields created by common current geometries.

The Right-Hand Grip Rule (for field direction)

Since Fleming's rule only gives the force, we use the Right-Hand Grip Rule (or Right-Hand Curl Rule) to determine the direction of the magnetic field ($B$) generated by a current ($I$):

Hold the wire/solenoid in your right hand:

• Your Thumb points in the direction of the conventional Current ($I$).
• Your Curled Fingers show the direction of the Magnetic Field Lines ($B$).

Field Patterns to Sketch and Describe

1. Long Straight Wire

   • Field lines are concentric circles centered on the wire.
   • Field strength decreases as distance from the wire increases.

2. Flat Circular Coil

   • The field looks like a straight wire nearby, but the circular loops combine.
   • At the centre of the coil, the field lines are straight and perpendicular to the plane of the coil.

3. Long Solenoid

   • A solenoid is a coil of wire acting like a cylindrical loop.
   • The magnetic field inside a long solenoid is uniform and parallel to the axis.
   • Outside, the field pattern closely resembles that of a bar magnet (North pole at one end, South pole at the other, determined by the Right-Hand Grip Rule).

Ferrous Cores

A ferrous core (a core made of iron or steel) inserted into a solenoid dramatically increases the strength of the magnetic field. This is because the core material itself becomes highly magnetized, adding its own field to the solenoid's field. This is the basis of electromagnets.

Forces Between Current-Carrying Conductors

Since a current-carrying wire creates a magnetic field, and another current-carrying wire experiences a force in a magnetic field, two parallel current-carrying wires will exert forces on each other.

• If the currents are flowing in the same direction, they attract.
• If the currents are flowing in opposite directions, they repel.

You can prove this using the Right-Hand Grip Rule (to find the $B$ field of wire 1) and then Fleming's Left-Hand Rule (to find the force on wire 2).

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20.5 Electromagnetic Induction

We have seen that electricity creates magnetism (current creates $B$). Now we look at the reverse: how magnetism creates electricity. This is called electromagnetic induction, the principle behind generators and transformers.

Magnetic Flux (\(\Phi\)) and Flux Linkage (\(N\Phi\))

Magnetic Flux (\(\Phi\)) is a measure of the total number of magnetic field lines passing through a given area.

\[\Phi = BA \cos \theta\]

For the simplest cases, where the area ($A$) is perpendicular to the flux density ($B$), \(\cos \theta = 1\), so:

\[\Phi = BA\]

• Unit of flux: Weber (Wb). (Since \(\Phi = BA\), 1 Wb = 1 T m\(^2\)).

Magnetic Flux Linkage (\(N\Phi\)) is the total magnetic flux passing through an entire coil of wire. If the coil has \(N\) turns, the flux linkage is:

\[\text{Flux Linkage} = N\Phi = NBA\]

Faraday's Law of Induction

Faraday’s Law tells us the magnitude of the e.m.f. induced in a circuit.

Statement: The magnitude of the induced electromotive force (e.m.f.) is directly proportional to the rate of change of magnetic flux linkage.

Mathematically:

\[E = - N \frac{\Delta \Phi}{\Delta t}\]

(The negative sign relates to Lenz's Law, covered below).

Factors Affecting the Induced E.M.F. (E)

To induce a larger e.m.f., you need a faster rate of change of flux linkage. This can be achieved by:

• Increasing $N$: Use a coil with more turns.
• Increasing $B$: Use a stronger magnet or field.
• Increasing $\Delta t$ (decreasing the time interval): Move the coil or magnet faster.
• Increasing $A$: Use a coil with a larger area.

The key is CHANGE. If the flux linkage is constant, the induced e.m.f. is zero.

Lenz's Law

While Faraday’s Law gives the size of the induced e.m.f., Lenz’s Law gives the direction. This law is essentially an application of the conservation of energy.

Statement: The induced e.m.f. acts in such a direction as to oppose the change in magnetic flux linkage that produced it.

Analogy: Lenz's Law is like a stubborn sibling. If you try to push a magnet's North pole into a coil, the coil immediately turns into a North pole to push back. If you try to pull it out, the coil becomes a South pole to pull it back in. It opposes any change to maintain the status quo.

How Lenz's Law Ensures Energy Conservation:

If the induced current enhanced the motion (rather than opposing it), the magnet would accelerate indefinitely, generating ever-increasing energy for free—violating the conservation of energy principle. You must always do work (apply a force over a distance) to overcome the opposing force described by Lenz's Law in order to generate electrical energy.