Modelling Magnetic Fields Revision Notes for HSC SSCE Physics

Modelling Magnetic Fields

Introduction

Magnetic fields can be represented and understood using different types of models. Each model has its own strengths and limitations, but all help us understand and predict magnetic behaviour. In physics, we use three main types of models: conceptual models, mathematical models, and numerical models.

Types of models

Field lines (conceptual model)

Field lines provide a visual way to represent and understand magnetic fields. Here's what field lines show us:

Direction: The arrows on field lines indicate the direction of the magnetic field at different points in space. Specifically, they show the direction of force that would act on a north magnetic pole.

Strength: The spacing between field lines indicates the strength of the magnetic field. Closer lines mean a stronger field, while lines farther apart indicate a weaker field.

infoNote

Important points about field lines:

Field lines don't physically exist - they are a modelling tool
You can choose how many lines to draw when creating a diagram
Field lines never show the path a particle would follow - they only show the direction of force at each point
Field lines never cross each other (because there can only be one direction of force at any point)

Field lines are used for electric, magnetic and gravitational fields. They help us visualize invisible forces and understand how fields behave in space.

Electron model (conceptual model)

The electron model helps us understand ferromagnetic materials (materials that can become permanent magnets). This conceptual model treats electrons as:

Charged particles moving in circular paths around atoms
Having their own magnetic field due to "spin"

This model explains how magnetic materials work:

In ferromagnetic materials, electrons' magnetic fields align within small regions called domains
When we apply an external magnetic field, it causes these domains to align
After the external field is removed, some domains stay aligned - the material is now magnetised

chatImportant

Limitations of this model:

It doesn't explain why electrons have their own magnetic field
It's a simplified representation, not the complete physical reality
More sophisticated models (quantum mechanics) are needed for deeper understanding

However, this model gives us useful explanatory and predictive power for understanding magnetic materials.

Mathematical models

Mathematical models use equations to describe magnetic fields precisely. They allow us to calculate specific values and make testable predictions.

Model 1: Long straight current-carrying wire

The magnetic field around a long straight wire carrying current $I$ is given by:

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

Where:

$B$ = magnetic field strength (in tesla, T)
$\mu_0$ = permeability of free space ( $4\pi \times 10^{-7}$ T·m/A)
$I$ = current through the wire (in amperes, A)
$r$ = distance from the wire (in metres, m)

chatImportant

Assumptions and limitations:

The wire must be very long compared to the distance $r$
The model works best when the measurement point is close to the wire relative to the wire's length
The model ignores effects near the ends of the wire
For short wires, this model becomes less accurate

When to use this model: Use it when the point where you're calculating the field is much closer to the wire than to either end of the wire.

Model 2: Current-carrying solenoid

The magnetic field inside a solenoid (coil of wire) carrying current is given by:

$B = \frac{\mu_0 NI}{L}$

Where:

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

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Assumptions and approximations:

The field is approximately constant throughout the inside of the solenoid
The solenoid is long compared to its radius
The wire is tightly wound (turns close together)

Note: The field is not perfectly uniform - it varies slightly near the ends of the solenoid, but is approximately constant in the middle section.

Numerical and computational models

What are analytical vs numerical solutions?

Analytical solutions involve solving an equation by performing mathematical operations to rearrange it until you have an expression for the variable you want. You then substitute values to calculate specific results.

Numerical solutions are used when equations are too difficult, time-consuming, or impossible to solve analytically. In these cases, we use computers to calculate many values. These are called numerical methods or computational simulations.

Why use numerical models?

Physicists use numerical models because:

Some equations cannot be solved analytically
We may need to calculate thousands of values (too time-consuming manually)
Complex systems require computer simulations
They allow us to explore "what if" scenarios quickly

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Numerical models can be created using:

Specialized mathematical software
Programming languages
Common spreadsheet software (for simpler simulations)

The advantage of numerical models is that they're based on mathematical models but use computers to do the calculations, allowing us to visualize and explore complex behaviour.

Investigation 14.7: Measuring the field inside a solenoid

Aim

To test the mathematical model for magnetic field inside a solenoid by comparing theoretical predictions with experimental measurements.

Hypothesis

Based on the equation $B = \frac{\mu_0 NI}{L}$ , the magnetic field inside the solenoid should:

Be proportional to the current $I$
Be approximately constant throughout the inside of the solenoid
Equal $\frac{\mu_0 N}{L}$ multiplied by the current

Materials

Hollow solenoid with known number of turns
Variable DC power supply
Ammeter
Magnetic field meter with probe

Safety considerations

Risk	How to manage
Electricity can cause shocks	Use only low voltages and currents

Consider other risks specific to your setup and how to manage them.

Method

Connect the solenoid and ammeter to the power supply in a series circuit.
Before turning on the power, measure the background magnetic field inside the solenoid using the field meter. Record the position of the probe - you must keep it in the same position for all measurements.
Turn on the power supply to a low voltage.
Record:
- The magnetic field inside the solenoid
- The current through the solenoid (from the ammeter)
Increase the voltage from the power supply.
Repeat steps 4 and 5 until you have at least six data points.
With current constant, slowly move the magnetic field meter probe around inside the solenoid. Note how the reading changes with position.

Results

Create a table with columns for current and magnetic field. Include:
- Appropriate units
- Uncertainties (check equipment manuals for precision)
Draw a large diagram of the solenoid showing how the field varied with position inside the coil, particularly near the ends.

Analysis of results

lightbulbExample

Analysis Steps:

Plot a graph of magnetic field (vertical axis) as a function of current (horizontal axis).
Draw a line of best fit and determine:
- The gradient (slope)
- The vertical axis intercept
Compare your results to the model $B = \frac{\mu_0 NI}{L}$ :
- Does the intercept match predictions?
- Does the gradient equal $\frac{\mu_0 N}{L}$ ?
How constant was the magnetic field inside the solenoid? Did it vary with position?

Conclusion

Write a conclusion that:

References your data and analysis
Evaluates how well the mathematical model matches your experimental results
Discusses any limitations or sources of error
Comments on the assumptions of the model (constant field, long solenoid)

Investigation 14.8: Modelling the magnetic field due to a current-carrying wire

Aim

To create a numerical model of the magnetic field around a long current-carrying wire using the equation $B = \frac{\mu_0 I}{2\pi r}$ and spreadsheet software.

Materials

Computer with spreadsheet software

Method

Setting up the spreadsheet

Open a new worksheet and add a title at the top. Save with a memorable filename.
Enter constant values near the top of the sheet:
- $\mu_0 = 1.2568 \times 10^{-6}$ T·m/A
- $I$ (choose a current value, e.g., 10 A)
- Add labels above each value
Create distance data ( $r$ $r$ ):
- Add a heading r (m) in a cell
- Enter your first distance value below (e.g., 0.001 m)
- In the cell below that, enter a formula to add a small increment (e.g., =A7+0.0005)
- Copy this formula down for about 100 cells
Calculate magnetic field data:
- Add heading B (T) next to the r column
- Enter a formula to calculate B using the equation (reference the cells containing $\mu_0$ and $I$ , and the r values)
- Use $ signs before cell references for constants (e.g., $A$ 4) so they don't change when copied
- Copy the formula down the same number of rows as your distance data

infoNote

When creating formulas in spreadsheets, use $ signs to create absolute cell references for constants. This ensures the reference doesn't change when you copy the formula to other cells.

Visualizing the results

Create a scatter plot:
- Plot magnetic field (vertical axis) versus distance (horizontal axis)
- Adjust axes for appropriate scales
- Add a descriptive title and axis labels
Explore your simulation:
- Change the current value and observe how the graph changes
- Save copies of graphs with different parameters
- Change the starting distance and increment values
- Observe how the magnetic field varies with distance

Results

Your results are the data tables and graphs you created.

Analysis of results

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Analyzing Your Numerical Model:

Does your graph show the expected relationship based on $B = \frac{\mu_0 I}{2\pi r}$ ?
- The field should decrease as distance increases
- The relationship should be inverse (hyperbolic curve)
What happens when you change:
- The current? (Field should be proportional to current)
- The starting distance? (Shows field at different ranges)
- The increment? (Changes the resolution of your model)
How could you extend this investigation?
- Model the field inside a solenoid using $B = \frac{\mu_0 NI}{L}$
- Compare fields from different currents on the same graph
- Calculate the field at specific distances

Conclusion

Summarize your findings about:

How well the numerical model represents the equation
The advantages of using spreadsheets for modelling
The limitations of this modelling approach

Understanding models and their limitations

All physics models have limitations because they make simplifying assumptions:

Field line model:

Limitations: Lines are arbitrary, don't physically exist, don't show particle paths
Strengths: Easy visualization, shows direction and relative strength

Electron model:

Limitations: Doesn't explain origin of electron magnetic field, simplified representation
Strengths: Helps understand magnetisation, domain alignment, ferromagnetic behaviour

Wire model ( $B = \frac{\mu_0 I}{2\pi r}$ ):

Limitations: Only works for very long wires, inaccurate near ends, can't calculate field inside wire ( $r = 0$ gives infinite field, which is unphysical)
Strengths: Simple equation, good predictions when assumptions are met

Solenoid model ( $B = \frac{\mu_0 NI}{L}$ ):

Limitations: Assumes constant field (actually varies near ends), requires long thin solenoid
Strengths: Simple equation, good approximation for centre region

Numerical models:

Limitations: Based on mathematical models (inherit their limitations), require computers
Strengths: Can solve complex equations, explore many scenarios quickly, visualize results

chatImportant

When using any model in physics, always ask yourself:

What assumptions does this model make?
Are these assumptions reasonable for my situation?
What are the limitations of this model?
When would this model give inaccurate results?

Remember!

bookmarkSummary

Key Points to Remember:

Magnetic fields can be modelled in three ways: field lines (conceptual), mathematical equations, and numerical simulations.
Field lines show direction and strength: Arrows indicate the direction of force on a north pole, and spacing indicates field strength. They don't physically exist but help us visualize fields.
Mathematical models have specific assumptions: The wire model assumes a very long wire; the solenoid model assumes a long, tightly-wound coil. Always check if assumptions are reasonable for your situation.
All models have limitations: Field lines are arbitrary, conceptual models are simplified, and mathematical models make approximations. Understanding these limitations helps you use models appropriately.
Numerical models use computers to solve equations: When equations are too complex for analytical solutions, computational simulations using spreadsheets or specialized software can calculate and visualize results.

Modelling Magnetic Fields (HSC SSCE Physics): Revision Notes