7.9.6 Required Practical: Investigating Flux Linkage on a Search Coil

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
  • This is just one example of how this required practical might be carried out

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 above. 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 (and 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 induced in the search coil is given by the expression:

• This is obtained through the following logic:
  • If a constant uniform magnetic field was used in this experiment, with a DC source, the flux linkage through the search coil would be given by the equation:
    • 

• • The magnetic field, however, is alternating with angular frequency ω
    • This is a result of the AC source used that oscillates with frequency f
    • Where ω = 2πf
  • As a result of this, the flux linkage through the coil oscillates at the same frequency as the AC source,
    • Current through the coil is given by: 
      • for peak current 
    • So, magnetic flux density is proportional to current and is given by:
      • for peak magnetic flux density 
    • Hence, 

• • To find the peak induced e.m.f in the search coil, apply Faraday's law (this is really just differentiating the above expression with respect to time):
    • 
• The expression for peak e.m.f, ε₀ , from the above equation is found when sin(ωt) is 1 or −1:
  • • 
  • Recall that we are able to measure ε₀  by halving the peak-to-peak voltage on the CRO
• Compare this to the straight-line equation: y = mx  + c
  • y = ε₀
  • m = B₀ANω
  • x = cos(θ )
  • c = 0

• Plot a graph of peak e.m.f ε₀ against cos(θ ) and draw a line of best fit
• This should be a straight-line graph
  • This shows that the induced e.m.f in the search coil 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 likely to have a y-intercept for cos(θ ) = 0

• 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 by fixed increments
• 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