We can see that the magnetic field strength is uniform within the solenoid. However the flux density becomes less at the ends, as the field lines get spread out.
We need a term that tells us the number of field lines, and it is called the magnetic flux. It is given the physics code F (‘Phi’, a Greek capital letter ‘Ph’), and has the units Weber (Wb). The formal definition is:
The product between the magnetic flux density and the area when the field is at right angles to the area.
In code we write:
F = BA
Remember that flux density is the number of field line per unit area, not unit volume!
The flux linkage is the flux multiplied by the number of turns of wire. If each turn cuts (or links) flux F, the total flux linkage for N turns must be NF. We can also write this as NBA. In other words:
Flux linkage = number of turns of wire ´ magnetic field strength ´ area
| How much flux links a 200 turn coil of area 0.1 m2 when it is placed at 90o to a magnetic field of strength 2.5 ´ 10-3 T? |
The flux linkage can be changed in two ways:
- We can alter the strength of the magnetic field;
- We can alter the area at 90o to the magnetic field by moving the coil. If we are turning the coil, the new flux linkage is given by NBA sin q where q is the angle the area makes to the magnetic field. When we move a coil across a magnetic field, the area swept is the change in area (just like the change in distance is the distance moved).
We give the change in flux linkage the physics code DF.
| The coil in question 4 is now turned so that it makes an angle of 60 o with the magnetic field lines. What is the change in flux linkage? |
Electromagnetic Induction
If we pass a current in a wire in a magnetic field, we know that the wire will move. It is therefore reasonable to suppose that if we move the wire in a magnetic field, and the wire is connected to an outside circuit, a voltage and current are induced. If the wire is not connected, a voltage only is induced. Consider this demonstration:
If we move the magnet parallel to the wire, the galvanometer hardly responds. However, if we move the magnet across the wire, then we see a definite reading on the galvanometer. The current (and voltage) induced on a single wire is rather small, but is increased by having more turns of wire. For any voltage to be induced, we must move the magnet. We call this voltage the induced electromotive force. It is often given the code Î, a fancy letter ‘E’.
Faraday’s Law and Lenz’s Law are two important rules that govern this effect.
Faraday’s Law is a formal definition of the effect:
The induced e.m.f. across a conductor is equal to the rate at which flux is cut.
Lenz’s Law says:
The direction of any induced current is such as to oppose the flux change that caused it.
The induced e.m.f. sets up a current that would oppose the force that is pulling the wire. If the force were to assist the motion, we would get acceleration, and an increase in kinetic energy. This would break the Law of Conservation of Energy. In other words, we cannot get something for nothing.
Lenz’s Law is important in motors and generators. As a motor speeds up, it acts like a generator to produce a back e.m.f. to oppose the current flowing in the motor. Therefore the current through a fast running motor is quite small. When it is running slowly, a big current flows. Electric motors are therefore very suited to railway use, where big currents are needed to get trains moving, and there is no need for a gearbox that is needed for a diesel engine.
The effect that we have seen is summed up in the relationship:
[N - number of turns; Î – e.m.f (V); DF/Dt – rate of change in flux (Wb/s)]
Worked Example
| A single turn of wire of cross-sectional area 5.0 cm2 is at 90o to a magnetic field of 0.02 T, which is reduced to 0 in 10 s at a steady rate. What is the e.m.f. induced? |
| Two formulae to use: F = BA and Î = - NDF Dt |
| We need to work out the flux: F = BA = 0.02 T × 5 × 10-4 m2 = 1 ×10-5 Wb |
| Now we can work out the e.m.f: Î = - NDF = 1× 1 ×10-5 Wb = 1 ×10-5 V Dt 10 s |
| A search coil has 2500 turns and an area of 1.5 ´ 10-4 m2. It is placed between the poles of a large horseshoe magnet. It is rapidly pulled out of the field in a time of 0.30 s. A datalogger records an average value for the emf of 0.75 V. What is the flux density between the poles of the magnet? |
We can use a magnetic field to induce a voltage in two ways:
1. Relative movement. The size of the voltage depends on:
- Speed the magnet passes through a coil or vice versa.
- Number of turns in the coil.
- Strength of the magnet.
2. Changing a magnetic field. We don’t have to make the magnetic field move. If we turn the current on or off, there is a change in the magnetic field, and that induces a voltage in a second unconnected coil. This is called the transformer effect or mutual induction.
Summary
Force on a current carrying conductor:
F = BIl
Force on a charged particle:
F = Bqv
Trajectory of a Charged Particle in a magnetic field:
v = Bqr
m
Flux Density
Magnetic field strength, B. “Concentration of field lines”
Flux
Product of field strength and area. F = BA. Total number of field lines.
Flux Linkage
Product of flux and the number of turns
Faraday’s and Lenz’s Laws
Î = - NDF
Dt
0 Comments to "Magnetic effect of current carrying conductor"
Leave a Reply