6.6.1 Magnetic Flux

• Magnetic flux is a quantity which signifies how much of a magnetic field passes perpendicularly through an area
• For example, the amount of magnetic flux through a rotating coil will vary as the coil rotates in the magnetic field
  • It is a maximum when the magnetic field lines are perpendicular to the coil area
  • It is at a minimum when the magnetic field lines are parallel to the coil area

• The magnetic flux is defined as:

The product of the magnetic flux density and the cross-sectional area perpendicular to the direction of the magnetic flux density

• Magnetic flux is defined by the symbol Φ (greek letter ‘phi’)
• It is measured in units of Webers (Wb)
• Magnetic flux can be calculated using the equation:

Φ = BA

• Where:
  • Φ = magnetic flux (Wb)
  • B = magnetic flux density (T)
  • A = cross-sectional area (m²)

The magnetic flux is maximised when the magnetic field lines and the area through which they are passing through are perpendicular

• When magnetic flux is not completely perpendicular to the area A, then the component of magnetic flux density B perpendicular to the area is taken

• The equation then becomes:

Φ = BA cos(θ)

• Where:
  • θ = angle between magnetic field lines and the line perpendicular to the plane of the area (often called the normal line) (degrees)

The magnetic flux decreases as the angle between the field lines and plane decrease

• This means the magnetic flux is:
  • Maximum = BA when cos(θ) =1 therefore θ = 0^o. The magnetic field lines are perpendicular to the plane of the area
  • Minimum = 0 when cos(θ) = 0 therefore θ = 90^o. The magnetic fields lines are parallel to the plane of the area

• An e.m.f is induced in a circuit when the magnetic flux linkage changes with respect to time
• This means an e.m.f is induced when there is:
  • A changing magnetic flux density B
  • A changing cross-sectional area A
  • A change in angle θ

Part (a)

Step 1: Write out the known quantities

• • Cross-sectional area, A = 40 cm × 73 cm = (40 × 10⁻²) × (73 × 10⁻²) = 0.292 m²
  • Magnetic flux density, B = 1.8 × 10⁻⁵ T

Step 2: Write down the equation for magnetic flux

Φ = BA

Step 3: Substitute in values

Φ = (1.8 × 10⁻⁵) × 0.292 = 5.256 × 10⁻⁶ = 5.3 × 10⁻⁶ Wb

Part (b)

• • The magnetic flux will be at a minimum when the window is opened by 90°
  • The magnetic flux will be at a maximum when fully closed or opened to 180°

Aims of the Experiment

• The overall aim of this experiment is to determine how the magnetic flux linkage varies with the angle of rotation of a search coil
  • This is done by rotating a search coil through a uniform magnetic field created by a larger coil and recording the induced e.m.f. within it

Variables

• Independent variable = Angle between the normal to the search coil and the magnetic field lines, θ
• Dependent variable= Induced e.m.f, ε
• Control variables:
  • Area of the search coil, A
  • Number of loops on both coils, N
  • Magnetic field strength, B
  • Frequency of the power supply, f

Equipment List

• Resolution of measuring equipment:
  • Protractor = 1º
  • CRO = 2 mV / div

Method

1. Arrange the apparatus as shown in the diagram.
   • The slinky spring should be connected to the alternating power supply so the flux through the search coil placed within it will be constantly changing
2. Set up the CRO so its time-base is switched off, so it only shows the amplitude of the e.m.f.
   • Adjust the voltage per division till the signal can be seen fully on the screen (eg. 10 mV / div)
3. Position the search coil so that it is halfway along the slinky spring
4. Orient the search coil so it is parallel to the slinky spring
   • The plane of its area is perpendicular to the field)
5. Record the induced e.m.f. in the search coil from the amplitude of the CRO trace
   • This should ideally be the peak-to-peak voltage (V_pp) which will then be halved for the peak e.m.f, ε₀
6. Rotate the search coil by 10º (in either direction) using the protractor
7. Record the new V_pp and repeat the procedure until the search coil is at 90º to the slinky spring

• An example table might look like this:

Analysing the Results

• The e.m.f. in the coil varies with the equation:

ε = BANω sin(θ)

• Where: 
• • ε = e.m.f. (V)
  • B = magnetic flux density (T)
  • A = cross-sectional area (m²)
  • N = number of turns
  • ω = angular velocity of the rotating coil (rad s⁻¹ or degrees  s⁻¹ )
  • θ = angle between the magnetic field, B and the normal to the area, A (rads)

• Comparing this to the straight-line equation: y = mx + c
  • y = ε
  • x = sin(θ)
  • m = BANω
  • c = 0

• Plot a graph of peak e.m.f. ε₀ against sin(θ) and draw a line of best fit
• This should be a straight-line graph
  • This shows that the induced e.m.f. is proportional to the cosine of the angle between the search coil and the direction of the magnetic field lines

Evaluating the Experiment

Systematic Errors:

• Reduce systematic errors by calibrating the search coil using a known magnetic field and oscilloscope
• The field lines are unlikely to be perfectly parallel and perpendicular to the area of the coil
  • Therefore, the graph is unlikely to have a y-intercept at the origin

• Read the angle from the protractor far above and from the same point every time to reduce parallax error

Random Errors:

• The experiment could be made more reliable by repeating for a full turn (θ = 360°)
• An improvement could be to use a calibrated motor to rotate the search coil at a steady rate
  • This will make the e.m.f. values more accurate
• Use blu tack to make sure the protractor stays in the same place for each reading

Safety Considerations

• Keep water or any fluids away from the electrical equipment
• Make sure no wires or connections are damaged and contain appropriate fuses to avoid a short circuit or a fire
• Don't exceed the specified current rating for the coil in order not to damage it
• The larger coil will heat up whilst the current is through it, especially if it is very thin
  • Therefore, make sure not to leave the current on for longer than necessary