2.2 ELECTROMAGNETISM 19th November 2012

Magnetic Fields A magnetic field exists around a moving charge in addition to its electric field. A current carrying conductor produces a circular field as shown below. The direction of the field is described by the left Handed Screw Rule, providing that we are dealing with electron flow current. If the thumb of the left hand points in the direction of current (electron flow) then the fingers show the field direction.

Magnetic Induction F = BIL The strength of a magnetic field is called the magnetic induction, B (or magnetic field density, or B-filed). It is measured in Tesla, T. The Tesla is defined as follows: “One Tesla is the magnetic induction in which a conductor of length one metre, carrying a current of one Ampere, perpendicular to the field, is acted on by a force of one Newton” A charged particle moving across a magnetic field experiences a force. The magnitude of the force depends on the magnetic induction, B, the current flowing (in the case of a current carrying conductor), I, and the length of the conductor perpendicular to the field. If the conductor lies perpendicular to the field then the force is given by: F = BIL

Table 4.2: Typical magnetic field values Situation Magnetic field (T) Magnetic field of the Earth 5 x 10-5 At the poles of a typical fridge magnet 1 x 10-3 Between the poles of a large electromagnet 1.00 In the interior of an atom 10.0 Largest steady field produced in a laboratory 45.0 At the surface of a neutron star (estimated) 1.0 x 108

In the general case where the conductor lies at an angle to the B-Field as shown: Θ I LsinΘ LsinΘ Θ LcosΘ F = BILsin Θ

Direction of force The direction of the force is given by the right hand rule. First Finger = Magnetic Field from North to South Thumb = force / motion SeCond Finger = CURRENT (electron flow)

Example Calculate the force on a wire of length 20cm at an angle of 30o to a magnetic induction of 12T, if it is carrying a current of 3A. F = BILsinΘ = 20 x 3 x 0.2 x 0.5 = 3.6N L = 0.2m Θ = 30o B = 12T F = 3A

Measuring Magnetic Induction A known length of wire is placed perpendicular to the magnetic induction between two permanent magnets on a sensitive balance. When a current is passed through the wire it experiences a force. Similarly, the wire exerts a force on the magnetic field which causes the reading on the balance to change. If the direction of the current is reversed then the sign of the reading, (Δm) also reverses. We can convert Δm into force, and by measuring the current, calculate B using: The direction of B can be found from the right hand rule.

Force Between Current Carrying Wires The magnetic induction around a current carrying wire has the shape shown below. . x Magnetic field due to a electron current travelling into the paper. Magnetic field due to a electron current travelling out of the paper. Remember that the direction of the magnetic field is obtained using the left handed screw rule for (electron flow) current.

Where, μ0 is the permeability of free space, 4π x10-7 TmA-1 In general, the magnetic induction, B, at a distance, r, from an infinite straight conductor carrying a current, I, is given by: Where, μ0 is the permeability of free space, 4π x10-7 TmA-1 Example Calculate the magnetic induction 220cm from a long straight wire carrying a current of 3A B = ? I = 3A r = 2.2m

Consider two parallel wires of infinite length separated by a distance r and carrying currents I1 and I2 in the same direction. If the wires are separated by a distance r, the magnetic induction at wire 2 due to the current in wire 1 is: So wire 2, carrying a current I2 , will experience a force (along length, L) F2 = B1I2L Substitute for B1 Similarly, wire 1 experiences a force due to the magnetic induction around wire 2 is known as the force per unit length This is contents statement 8 DERIVE

Note: For wires carrying current in the same direction, the forces are attractive. For wires carrying currents in opposite directions, the forces are repulsive. Example Two long parallel wires are 5cm apart. They exert a force per unit length of 6x10-7Nm-1 on each other. If one wire carries a current of 400mA, calculate the current in the second wire. r = 0.05m I1 = 0.4A I2= ? F/L = 6x10-7Nm-1 Is known as the force per unit length

Definition of the Ampere A current of one ampere is defined as the constant current which, if in two straight parallel conductors of infinite length placed one metre apart in a vacuum, will produce a force between the conductors of 2 x 10-7 Newton's per metre. To confirm this definition apply to this situation. Thus I1 and I2 both equal 1 A, r is 1 m and μo = 4π x 10-7 N A-2.