# Magnetic Flux Density ## What is Magnetic Flux Density? - Magnetic flux density $B$ describes the strength of a magnetic field - Also referred to as **magnetic field strength** - A fundamental quantity used to calculate magnetic forces

Magnetic flux density is the force acting per unit current per unit length on a current-carrying conductor placed perpendicular to the magnetic field

## Defining Equation - Rearranging the magnetic force equation gives the definition of magnetic flux density: $$B = \frac{F}{IL}$$ - Where: - $B$ = magnetic flux density (T) - $F$ = magnetic force (N) - $I$ = current (A) - $L$ = length of the wire (m) - **Important**: This equation only applies when the B-field is **perpendicular** to the current - If the conductor is at an angle $\theta$ to the field, use: $B = \frac{F}{IL\sin\theta}$ ## The Tesla - Magnetic flux density is measured in **tesla** (T) - One tesla is defined as: **A wire carrying a current of 1 A normal to a magnetic field of flux density 1 T experiences a force per unit length of the conductor of 1 N m$^{-1}$** - To put this in perspective: - Earth's magnetic flux density: ~0.032 mT - Ordinary fridge magnet: ~5 mT - Laboratory electromagnets can reach several tesla ## Experimental Determination of Magnetic Flux Density ### Aim - To calculate the magnetic flux density of a magnet - This is done by measuring the force on a current-carrying wire placed perpendicular to the field ### Variables - **Independent variable**: Current, $I$ - **Dependent variable**: Mass, $m$ (which relates to force) - **Control variables**: - Length of wire, $L$ - Magnetic flux density, $B$ - Potential difference of the power supply ### Equipment | **Apparatus** | **Purpose** | |--------------|-------------| | Electronic top-pan balance | To measure the mass | | Thick copper wire | The object which experiences the magnetic force | | Variable resistor | To vary the current flowing through the wire | | DC power supply | To provide the potential difference through the wire | | Ammeter | To measure the current through the wire | | 2 magnets with opposite poles in a metal cradle | To create the magnetic field | | Clamp (retort) stand | To hold the wire in the magnetic field | | 30 cm ruler | To measure the length of the magnet (length of wire in field) | | Crocodile clips | To hold both ends of the wire in place on the clamp stand | - **Resolution** of measuring equipment: - Ammeter = 0.01 A - Variable resistor = 0.01 Ω - Top-pan balance = 0.01 g - Ruler = 1 mm ### Method 1. Set up the apparatus as shown: - Wire positioned completely perpendicular between the magnets - Magnets placed in a metal cradle on the top-pan balance - Wire clamped securely above the magnets 2. Measure the length of one of the magnets using the 30 cm ruler - This is the length of wire $L$ in the magnetic field 3. With no current flowing, reset the top-pan balance to 0 g - This zeroes out the weight of the magnets and cradle 4. Adjust the variable resistor so a current of 0.5 A flows through the wire (as measured on the ammeter) 5. The wire experiences a force upwards - By Newton's third law, an equal force pushes downwards on the magnets - This registers as additional mass on the balance - The movement is very small and may not be visible 6. Record the mass on the top-pan balance for this current value 7. Increase the current in intervals of 0.5 A - Take 8-10 readings total (not exceeding 6 A to avoid overheating) 8. Repeat the experiment at least 3 times - Calculate the mean of the mass readings for each current value ### Example Results Table | **Current** $I$/A | **Mass** $m_1$/g | **Mass** $m_2$/g | **Mass** $m_3$/g | **Mean Mass** $m$/g | |-------------------|------------------|------------------|------------------|---------------------| | 0.50 | | | | | | 1.00 | | | | | | 1.50 | | | | | | 2.00 | | | | | | 2.50 | | | | | | 3.00 | | | | | | 3.50 | | | | | | 4.00 | | | | | | 4.50 | | | | | | 5.00 | | | | | ### Analysis - The magnetic force on the wire is: $$F = BIL$$ - Since $F = mg$ (where $m$ is in kilograms), we can equate: $$mg = BIL$$ - Rearranging for mass $m$: $$m = \frac{BL}{g} \times I$$ - Comparing to the straight-line equation $y = mx + c$: - $y = m$ (mass) - $x = I$ (current) - gradient $= \frac{BL}{g}$ - $c = 0$ (y-intercept is zero) **To find magnetic flux density $B$:** 1. Plot a graph of mean mass $m$ (y-axis) against current $I$ (x-axis) 2. Draw a line of best fit 3. Calculate the gradient of the line 4. Use the equation: $$B = \frac{g \times \text{gradient}}{L}$$ - Where: - $g = 9.81$ m s$^{-2}$ - gradient = gradient of the graph (in kg A$^{-1}$ or $\times 10^{-3}$ kg A$^{-1}$ if mass in grams) - $L$ = length of wire in the magnetic field (m) **Example graph showing the linear relationship between current and mass:**

The graph should pass through the origin - if it doesn't, check for zero errors on the balance

### Evaluating the Experiment **Systematic Errors:** - Ensure the top-pan balance starts at 0 g to avoid a zero error - Zero the balance **after** placing the magnets but **before** passing current **Random Errors:** - Repeat the experiment by rotating the magnet in the metal cradle and the wire by 90° - Take multiple readings and calculate a mean - Ensure the wire remains perpendicular to the magnetic field throughout **Safety Considerations:** - Keep water and fluids away from electrical equipment - Check all wires and connections are undamaged - Use appropriate fuses to prevent short circuits or fire - High currents cause the wire to heat up - do not touch the wire when current is flowing - Do not exceed 6 A to prevent overheating and changes in resistance ## Worked Example: Calculating B from Force **Question**: A 15 cm length of wire is placed vertically and at right angles to a magnetic field. When a current of 3.0 A flows vertically upwards, a force of 0.04 N acts on it to the left. Determine the flux density of the field and its direction. **Answer:** **Step 1**: Write out known quantities - Force on wire, $F = 0.04$ N - Current, $I = 3.0$ A - Length of wire, $L = 15$ cm $= 15 \times 10^{-2}$ m **Step 2**: Write the magnetic flux density equation $$B = \frac{F}{IL}$$ **Step 3**: Substitute values $$B = \frac{0.04}{3 \times (15 \times 10^{-2})} = 0.09 \text{ T}$$ **Step 4**: Determine direction using Fleming's left-hand rule - $F$ = to the left - $I$ = vertically upwards (conventional current) - Therefore, $B$ = **out of the page** ## Worked Example: Experimental Data Analysis **Question**: A student investigates the relationship between current and mass produced from the magnetic force on a current-carrying wire. They obtain results showing mass increases linearly with current from 0.11 g at 0.50 A to 1.10 g at 5.00 A. The mean length of the wire in the magnetic field was 0.05 m. Calculate the magnetic flux density of the magnets. **Answer:** **Step 1**: Complete the data analysis - Need to plot mass against current and find the gradient **Step 2**: Calculate the gradient - From the graph: gradient $= \frac{(1.0 - 0.2) \times 10^{-3}}{4.625 - 0.875} = 0.2133 \times 10^{-3}$ kg A$^{-1}$ **Step 3**: Calculate magnetic flux density $$B = \frac{g \times \text{gradient}}{L}$$ $$B = \frac{9.81 \times (0.21333 \times 10^{-3})}{0.05} = 0.042 \text{ T} = 42 \text{ mT}$$

Always convert mass to kilograms before calculating - common mistake is to leave mass in grams!

## Key Points to Remember - Magnetic flux density $B$ quantifies the strength of a magnetic field - The tesla is defined in terms of force per unit current per unit length - $B$ can be determined experimentally using a current-carrying wire and digital balance - The relationship $m = \frac{BL}{g}I$ gives a linear graph passing through the origin - The gradient of a mass vs current graph allows calculation of $B$ - Always ensure the wire is perpendicular to the field for maximum force