Electromagnetism
- Unlike pole attract each other while like poles repel each other.
- They exist in pairs, the magnets contain N and S pole at the same time while magnetic monopole does not exist in classical physics.
- The magnetic field points from N to S, it's the direction of force acting on a hypothesized N monopole.
- Magnetic field density B, often referred as magnetic field, has unit Tesla (T). It's proportional to the field lines on the diagram.
They are measured by two instruments:
1) Hall probe for fixed B; 2) search coil for varying B over time
3) Test current balance and 4) Other coil in various set up
When current passes through a conductor, magnetic field is induced following the right hand grip rule. A current out of paper would have anti-clockwise magnetic field, and opposite current gives opposite B direction. The magnetic field due to current in conductor is given by B = μ_{0}I/2πr, where I is the current, r is the distance from the conductor, μ_{0} = 4π*10^{-7} Tm/A. is the magnetic constant B is a vector and can be added through vector sum.
When we bend the conductor into a series of circle, which is called solenoid, we have a uniform magnetic field inside the solenoid. B inside the solenoid is given by B = μ_{0}nI = μ_{0}IN/L. Note that n=N/L, N is the turns of solenoid and L is its length. It's noted that radius of solenoid r<<L.
At two ends of the solenoid, B is just half as that of inside the solenoid, and out of the solenoid, B decreases gradually to zero due to flux leakage.
When a segment of conducting wire is placed in the uniform magnetic field, the force experienced is given by F = L(B x I) (the cross product) = BILsinθ. The direction of F can be given by the Fleming's left hand rule.
Consider the force per unit length between two conducting wire, assume they are parallel:
F/ L = BI'L = (μ_{0}I/2πr )I' = μ_{0}II'/2πr, it follows that when the direction of current are the same, the force is attractive.
1 A can be defined as the current which flowing in each of the two infinitely long parallel straight conductors of negligibly small cross-sectional area, separated 1 m in vacuum, exerting 2*10^{-7} Nm^{-1} on each other.
In a current balance, there's a rectangular conducting loop, but one side is insulated in order to balance the weight of the loop. Place a pair of slab magnet on the conducting side so that there's upward force acting on the conductor. By putting extra rider on the loop we are able to calculate the magnitude of magnetic force, then we can calculate the B field strength.
Motor
1) The original motor
As in the figure, it rotates due to the moment acting on the coil. However it tends to stop at perpendicular state since the force is balanced and when the coil turns, the force acting on the coil tends to push back the coil to the perpendicular state.
2) Improved motor
A commutator is applied so that the direction of current in the coil is inversed when the coil passes through the perpendicular axis of coil. Then the coil spins simultaneously. The disadvantage is that the angular acceleration is unstable, and then its speed is unstable.
Consider the couple along the spinning axis,
τ = 2Fs = 2(BIL)(w/2 sinθ) = NBAI sinθ, where w is the width of coil, A is the area of coil, θ is the angle between coil and vertical plane. It follows that F = 0 when it's perpendicular. It spins due to inertia.
3) Practical motor
We use concave cylindrical magnet such that a radial magnetic field is set up. Then B perpendicular to I, τ = NBAI, then its acceleration is stable.
In practical, there're three pairs of coil in the ammature and soft iron core inside it.
We can increase the magnitude of force by:
- Increase number of loops of coil, intensify B field or current passing through.
- Lengthen the magnet (if it's shorter than the coil), add soft iron core or lengthen the width.
These method can be proved by τ = NBAI or the concept of magnetic flux.
Motion of charged particles in a magnetic field
Charge carrier is a microscopic view on conductors:
- Electrolyte: positive and negative ions
- Vacuum tube: electron, discharge tube: electron and positive ions
- Metal: free electron (delocalized)
- N-type semiconductor: positive holes; P-type semiconductors: free electrons
Definition of current: I = dQ/dt = neva (A generalized form nqva, where n is the number charge carrier per unit volume, q is the charge per unit carrier, v is the drift velocity, a is the cross-sectional area of conductors. When the charge carrier is electron we write as neva for simplicity.) It refers to charge passing through a point in a unit of time.
Drift velocity is the net movement of electron. Due to thermal motion it has the speed of 10^{6}ms^{-1}, but the net movement is zero. Under p.d. or electric field, it has a net movement of about 10^{-4}ms^{-1}, which causes the current to occur. At the same time, the drift velocity along the wire is unique since once the circuit is open, electrical signal transmits about the speed of light so that all charge carriers stop drifting at the same time.
When a magnetic force is produced on a conducting wire, microscopically we say magnetic force is acting on the charge carriers. Assume N is the number of charge carriers in a piece of conductor, by n = N/AL, I = nqvA = (N/AL)qvA = Nqv/L, force acting on the conductor = force acting on all the charge carriers = BILsinθ = B(Nqv/L)Lsinθ = N(qvBsinθ)
Therefore the magnetic force acting on a single charge carrier F = qvBsinθ, or F = q(v x B).
1) When v is perpendicular to B
In this case, F = qvB which is perpendicular to v and B, then the charge performs a circular motion until it goes out of the magnetic field in opposite direction and same speed.
2) When v is parallel to B
In this case, F = 0, the charge carrier is not affected by the B field, so it undergoes rectilinear motion.
3) Other case
We put the velocity vector into two components: perpendicular and parallel to the B field. The perpendicular sector is affected by B field so that it undergoes circular motion, at the same time the parallel component is unaffected, combining gives a helical motion.
Hall voltage: the phenomenon of generating voltage under magnetic field in direction perpendicular to the flow of current.
Consider the following figure. Due to magnetic force, the charge carriers are deflected and accumulate in the two sides of the conductor so that one side is relatively positively charged and another side is relatively negatively charged. Then a electric field is set up between the two plates and the electric force acting on the charge carriers is opposite to the magnetic force. When F_{E}=F_{M}, a steady p.d. is set up between the two plates.
Note that the polarity of two plates are the same no matter the charge carrier is positive or negative since when the polarity of charge reversed, force acting on it and the deflecting direction is reversed. Positive charge going to the left is same as negative charge going to the right.
Balancing the force: qE = qV/d = qvB, E = V_{H}/d = vB = (I/nqA)B = IB/nqdt, we have V_{H} = IB/nqt, where V_{H} is the hall voltage and t is the thickness of conductor (the dimension that parallels B)
It can be used to measure the magnitude of magnetic field. In order to raise the sensitivity, semiconductor is used so that n is much smaller, while V_{H} is still measured in mV. The apparatus is called Hall Probe, where a semiconductor is connected with a d.c. supply, and vertically connected with a millivoltmeter, then voltage is registered when B field is passing through (perpendicular to I and B). Besides, the connection of millivoltmeter may not be accurate so potentiometer is used to offset the unexpected p.d.. It's connected across the semiconductor.
Precautions when measuring B-field
- Make sure the orientation is correct. For example, in hall probe, adjust its direction so that B field to be measured is perpendicular to I and B.
- Measure in the east-west orientation if possible so that the B field due to the Earth (N-S) does not affect your result.
Some applications on electromagnetism: maglev train, speakerphone, mass spectrometer, speed selector.
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