Magnetic flux density

​​In a nutshell

When a current carrying conductor is placed inside a magnetic field, the two magnets experience an equal and opposite force of which the direction can be determined with Fleming's left-hand rule. The magnetic flux density is defined as the force per unit current per unit length acting on a wire at right angles to a magnetic field.
​

Equations

Variable definitions

Force on a current carrying conductor

When current passes through a conductor it produces its own magnetic field. If you place it inside another magnetic field, the two magnets will interact and experience an equal and opposite force. This effect on a current carrying wire is shown below:

1. Current 2. Magnetic field 3. Force

The direction of the force can be determined using Fleming's left-hand rule, shown below:

1. Current 2. Magnetic field 3. Force

Note: The direction of the force will also be the direction of the motion. 

​

Magnetic flux density

The magnitude of the force on the wire depends on the strength of the field, the current of the wire, the length of the wire that is inside the field and the angle between the wire and the field. It can be calculated using the equation:

​$$F=BILsin\theta$$​​

When the field is at right angles to the wire, $$sin(90)=1$$, hence the equation becomes:

​$$F=BIL$$​​

By rearranging this equation one can find ​$$B$$, a quantity known as the magnetic flux density which is measured in tesla $$T$$. It is defined as the force per unit current per unit length acting on a wire at right angles to a magnetic field. You can think of it as the strength of the magnetic field:

$$B=\dfrac{F}{IL}$$​​

Example

A $$20\ cm$$ wire is placed perpendicularly to a magnetic field with a magnetic flux density of $$3\ T$$. What is the magnitude of the force acting on the wire if a current of $$0.5\ A$$passes through it?

Firstly, write down what you know:

$$L=20\ cm=0.2\ m\newline B=3\ T\newline I=0.5\ A$$​​

Next, write down the equation for the force on a wire in a magnetic field:

​$$F=BILsin\theta$$​

Since the wire is at right angles to the field the equation becomes:

$$F=BIL$$​​

Substitute the numbers into the equation and calculate the force on the wire:

$$F=3\times0.5\times0.2=0.3\ N$$​​

The magnitude of the force acting on the wire is $$\underline{0.3\ N}$$​.

Determining magnetic flux density

To determine the magnetic flux density between two magnets, you can set up an apparatus as shown below:

1. Crocodile clip 2. Power source 3. Arrangement 4. Clamp 5. Wire 6. Magnets

The idea is that when the circuit is on, the force will cause a weight difference in the magnets which will be picked up by the top-pan balance. You can then work out the force using the equation:

​$$F=mg$$​​

​

Where $$g$$​ is the acceleration of free fall $$9.81\ m\ s^{-2}$$. Using the length of the magnets and the current in the wire, you can find the magnetic flux density through the equation:

$$B=\dfrac{F}{IL}$$​​