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Measuring the Strength of a Magnetic Field
When a current flows through a conductor which is in a magnetic field, the conductor experiences a force, except when the current flows parallel to the magnetic field lines.
The direction of the force is at 90 to both the current and the field.
 
John Ambrose Fleming noticed a way to remember the relation between the three directions, of current, field and force (or resulting motion), involved here.
Place the thumb, first finger and second finger of your left hand at 90 to each other (or as near as you can manage!)
Suppose you want to find the direction (and sense) of the force on a conductor in a magnetic field.
Place your First finger in the direction of the Field, seCond finger in the direction of the Current (conventional current, of course) and you will then find that your thuMb indicates the direction of the force (or resulting Motion).
This is now called Fleming's Left Hand Rule (can you guess why?)
 
The effect can be observed using apparatus like that shown below.
 
A small moveable copper tube makes contact with two fixed copper "rails" along which it can roll.
When current is passed through the system, a shown, the copper tube moves to the right or left depending on the orientation of the magnet and the sense of the current. (In the situation shown above it moves to the right.)
 
Sometimes you might need to stand on your head to get your hand in approximately the same position as the apparatus being investigated but, let's face it, that's a small price to pay for scientific progress.
 
 
Experiments show that the magnitude of the force acting on a current-carrying conductor in a magnetic field
- is directly proportional to the current, I
- is directly proportional to the length, L of the conductor in the field
- depends on the angle between the current and the field.
 
We therefore have
so
and if the angle between the current and the field is 90 then the constant is called the magnetic flux density (or magnetic field strength), symbol, B.
This means that we can measure the strength of a magnetic field by passing a known current through it, at 90 to the field, and measuring the force exerted on the conductor carrying the current.
The magnetic flux density of a magnetic field is defined as:
The force per unit length per unit current acting on a conductor placed at 90 to the field.
Therefore, the units of B are NA-1m-1 but 1NA-1m-1 is called 1Tesla (after Nicola Tesla)
 
If we have a situation in which the angle between the field and the current is not 90 but some other angle, θ then the force per unit current per unit length has a smaller magnitude. 
We know from experiments that the force is zero when the angle is zero and maximum when the angle is 90.
We might therefore guess that there will be a sinq in the equation somewhere.
Looking at this a little more carefully:
If the angle between the current and the field lines is q then the magnitude of the component of B which is at 90 is equal to Bcos(90-θ) which is equal to Bsinθ, see diagrams below.
In conclusion, for a conductor carrying a current, I at an angle θ to a magnetic field of flux density B, the force acting on the conductor is given by
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