Magnetic Flux Density

The force applied to the current-carrying wire is determined by the equation:

F = BIL

• F is the force on the wire in Newtons \left(\text{N}\right).
• B is the magnetic flux density in tesla \left(\text{T}\right).
• I is the current in amps \left(\text{A}\right).
• L is the length of the wire in metres\left(\text{m}\right).

It is important for the above equation to work that the current carrying wire and the magnetic field of the magnet are perpendicular to each-other.

If not, the equation can be adapted to:

F = BIL \sin \theta

• \theta is the angle between the wire and the magnetic field lines of the magnets.

Therefore, if the current-carrying wire and the magnetic field are parallel \left(\theta = 0 \degree\right), \sin \theta would equal zero and no force would be produced.

Example: A current of 2 \: \text{A} flows through a wire perpendicular to a magnetic field of flux density \left(B\right) 100 \: \text{mT}. The length of the wire is 0.5 \: \text{m}. What is the magnitude of force on the wire?

[2 marks]

Because the wire is perpendicular to the field, \sin \theta = \sin 90 = 1 .

F = BIL

F = 100 \times 10^{-3} \times 2 \times 0.5

F = 100 \times 10^{-3} \: \text{N}

We can see that the direction of the force \left(F\right), current \left(I\right) and Magnetic Flux \left(B\right) have been labelled. Fleming’s left hand rule can be used to determine the direction of these variables. As the name suggests, it is important that the left hand is used and never the right.

Next to the diagram is a representation of Fleming’s left hand rule. The thumb represents the force or motion, the first finger is the direction of the magnetic flux and the second finger represents the current in the wire. As long as two of the directions are known, the third can be found by rotating the left hand to match the directions of the two variables known.

 

Investigating how force varies with current through a wire in a magnetic field.

This experiment will use the following set up to observe how changing the current through a wire in a magnetic field will effect the force on this wire.

A similar experiment can be conducted where instead of the current through the wire being changed, the length of wire or strength of magnetic field is changed.

Doing the experiment:

1. Set up the experiment as shown above, with the power supply initially switched off. Place the magnet on a top pan balance. Reset the top pan balance to 0.0 \: \text{g}. Also measure the length of wire.
2. Ensure the wire is completely perpendicular to the magnets as this would influence the experiments result if not.
3. Measure the length of the magnets as this will give us the length of wire \left(L\right) inside the magnetic field.
4. Switch the power supply on and adjust the current to 0.5 \: \text{A} using the variable resistor.
5. As a result of the current now flowing through the wire, an upwards force is exerted on the wire. Due to Newton’s 3rd Law, there is equal and opposite force exerted on the magnet by the current carrying wire. Therefore there is a reading on the top pan balance. Record this reading on the top pan balance.
6. Repeat this method but each time increase the current in intervals of 0.5 \: \text{A}, up until an appropriate maximum current.
7. Repeat the entire experiment for each current 3 times and take a mean average mass for each current.

Analysing the results:

As the force on a current carrying wire is given by the equation;

F = BIL

The top pan balance measures the mass pushing down upon it. We know that the corresponding force is the mass multiplied by the gravitational acceleration: F = mg.

Therefore we can rewrite the equation as:

mg = BIL

So in order to calculate the magnetic flux density we can use the equation:

\text{gradient} = \dfrac{BL}{g}

B = \dfrac{\text{gradient} \times g}{L}