Physics (9630) | Magnetic flux density

Welcome to the World of Magnetic Fields!

Hello! This chapter takes us deeper into the fascinating world of magnetism, moving beyond just drawing field lines. We are going to learn how to quantify magnetic fields – that is, how to measure exactly how strong a magnetic field is. This strength is called Magnetic Flux Density, often represented by the symbol \(B\).

Why is this important? Understanding \(B\) is crucial because it governs the forces that make electric motors spin, televisions work, and particle accelerators guide subatomic particles. It’s the mathematics behind the magnetic forces you already know!

Quick Review: Magnetic Fields

A magnetic field is a region where a magnetic force can be experienced. Magnetic field lines point from North to South. We often use simple symbols to show field direction:

• \(\mathbf{X}\): Field lines going into the page (like the tail feathers of an arrow).
• \(\mathbf{\bullet}\): Field lines coming out of the page (like the tip of an arrow).

3.10.1 Magnetic Flux Density (\(B\)) and Force on a Wire

The most fundamental way to define magnetic flux density is by measuring the force it exerts on a current-carrying conductor.

Defining Magnetic Flux Density, \(B\)

When a wire carrying current (\(I\)) is placed inside a magnetic field (\(B\)), it experiences a force (\(F\)). If the wire is of length (\(L\)) and is placed perpendicular to the field lines, the relationship is:

$$F = BIL$$ \(F\): Force (N)
\(B\): Magnetic Flux Density (T)
\(I\): Current (A)
\(L\): Length of wire within the field (m)

If the wire is not perpendicular, the force is slightly more complex, but for this level, we focus on the case where the wire and field are at 90 degrees.

What is Magnetic Flux Density (\(B\))?

We can rearrange the formula to define \(B\):

$$B = \frac{F}{IL}$$

Magnetic flux density is therefore defined as the force per unit current per unit length acting on a conductor placed perpendicular to the field.

Key Term: The SI unit for Magnetic Flux Density (\(B\)) is the tesla (\(\mathbf{T}\)).

One Tesla (1 T) is the magnetic flux density that causes a force of 1 Newton (1 N) on a wire of length 1 metre (1 m) carrying a current of 1 Ampere (1 A), when the wire is perpendicular to the field.

Did you know? The Earth's magnetic field strength is typically around 0.00005 T, while the large electromagnets used in MRI scanners can produce fields of 3 T or more!

Fleming’s Left-Hand Rule (The Motor Rule)

Magnetic flux density (\(B\)) is a vector quantity (it has direction). The force, current, and field are all mutually perpendicular (at 90 degrees) to each other. To find the direction of the force, we use Fleming’s Left-Hand Rule.

Memory Aid: Use the F-B-I mnemonic with your left hand:

1. Forefinger: Direction of the Field (B).
2. Middle finger: Direction of the Current (I) – remember, current is flow of positive charge (Conventional Current).
3. Thumb: Direction of the Force (F) or motion.

Analogy: Think of your fingers telling the story of an electric motor! The field is the setup, the current is the input, and the force (motion) is the output.

3.10.2 Moving Charges in a Magnetic Field

Current is just the movement of charge. If a wire experiences a force, then the individual charges moving inside that wire must also experience a force!

Force on a Charged Particle, \(F\)

When a single charged particle (charge \(Q\)) moves at velocity (\(v\)) perpendicular to a magnetic field (\(B\)), the force it experiences is given by:

$$F = BQv$$ \(F\): Force (N)
\(B\): Magnetic Flux Density (T)
\(Q\): Charge of the particle (C)
\(v\): Velocity of the particle (m s\(\text{}^{-1}\))

Don't worry if this seems tricky! This formula is essentially derived directly from \(F=BIL\). It simply applies the concept to a microscopic level.

Direction of Force (Using Fleming's LHR)

We still use Fleming's Left-Hand Rule, but we need to remember one critical distinction:

• For Positive Charges (like protons): Use the Left-Hand Rule exactly as is. The Middle finger points in the direction of velocity (\(v\)).
• For Negative Charges (like electrons): The force acts in the direction opposite to that predicted by the Left-Hand Rule, because the current direction is opposite to the movement of negative charge.

Circular Path of Particles

If a charged particle enters a uniform magnetic field perpendicularly, the magnetic force (\(F = BQv\)) is always perpendicular to the velocity (\(v\)).

What happens when a force is always perpendicular to motion? The force doesn't change the speed of the particle; it only changes its direction. This causes the particle to move in a perfect circle.

In this case, the magnetic force provides the necessary centripetal force (\(F_{\text{c}}\)).

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

By equating these forces, we can find the radius (\(r\)) of the circular path:

$$r = \frac{mv}{BQ}$$

This principle is used in devices like the cyclotron, which accelerates charged particles by guiding them in increasingly wide circular paths using a strong magnetic field.

3.10.3 Magnetic Flux and Flux Linkage

So far, we have looked at \(B\) as a measure of force capability. Now, let’s look at it geometrically using the concept of flux.

Magnetic Flux, \(\Phi\)

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

Key Concept: Magnetic flux depends on both the strength of the field (\(B\)) and the size of the area (\(A\)) it passes through.

If the magnetic field \(B\) passes perpendicularly through the area \(A\):

$$ \Phi = BA $$

Key Term: The unit of magnetic flux (\(\Phi\)) is the weber (\(\mathbf{Wb}\)).

Since \(\Phi = BA\), the units are: \(1 \text{ Wb} = 1 \text{ T} \cdot \text{m}^2\). This means 1 Tesla is also defined as one Weber per square metre.

The Importance of Angle

What if the area is tilted? Imagine holding a solar panel. You want the maximum amount of sunlight (flux) to hit the panel. This only happens when the panel is perpendicular to the light rays.

Magnetic flux is maximum when the field lines are perpendicular to the area, and zero when the field lines are parallel to the area.

If the field \(B\) is at an angle \(\theta\) to the normal (a line perpendicular to the area):

$$ \Phi = BA \cos \theta $$

Note: In this equation, \(\theta\) is always measured between the magnetic field vector \(B\) and the normal to the coil's area \(A\).

• If \(\theta = 0^{\circ}\) (Normal parallel to B), \(\cos 0^{\circ} = 1\), Flux is maximum (\(\Phi = BA\)).
• If \(\theta = 90^{\circ}\) (Normal perpendicular to B, meaning the area is parallel to B), \(\cos 90^{\circ} = 0\), Flux is zero (\(\Phi = 0\)).

Magnetic Flux Linkage, \(N\Phi\)

In physics, especially when dealing with motors or generators, we usually work with coils of wire, which have multiple turns.

Magnetic Flux Linkage is simply the total magnetic flux passing through all the turns of the coil.

If a coil has \(N\) turns and each turn has flux \(\Phi\), the flux linkage is:

$$ \text{Flux Linkage} = N\Phi $$

For a rectangular coil of \(N\) turns, area \(A\), rotated in a field \(B\):

$$ N\Phi = BAN \cos \theta $$

This concept of flux linkage is vital because any change in flux linkage is what generates electricity (Electromagnetic Induction), which you will study next.