Magnetism

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Magnetic fields can be caused in three different ways1. A moving electrical charge such as a wire with current flowing in it2. By electrons or protons which act like little bar magnets3. By some magnetic atoms such as iron (ferromagnetism)

All of the appearances of magnetism can be attributed to the movement of electrical charges or the magnetic moment from elementary particles as a result of their spin. In order to describe the phenomena of magnetism, we need the concept of a magnetic field. Magnetic fields can be caused by1.magnetic material such as a permanent magnet 2.electric current such as current flowing through an inductor3. the change in time of an electric field.

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InductorMagnetic field caused by current flowing through some coils.

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Lorentz Force The magnetic field has an effect on the force of moving charges. This is called the Lorentz force. The Lorentz force is the force on a point charge due to electromagnetic fields. It takes effect perpendicular to the field lines of the magnetic field and perpendicular to the direction of the movement of the charge.

whereF is the force (in newtons) E is the electric field (in volts per metre) B is the magnetic field (in teslas) q is the electric charge of the particle (in coulombs) v is the instantaneous velocity of the particle (in metres per second)

Biot-Savart Law Consider a current I flowing through a wire. Break the wire intolittle pieces of length ds. The magnetic field due to this little piece of current is then found experimentally to be

or

This is the law of Biot-Savart. r is the distance from the current element I ds to the field point P where we wish to find the magnetic field B. is a unit vector pointing along r. is a constant of nature, the permeability of free space.

The relationship between ds, er, and dB is shown in the figure. To determine the direction of use your right hand and point your fingers along ds and curl them toward er. Your right thumb will point along dB.

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To find the total magnetic field due to a conductor, we add up the contributions from each current element by integrating over theentire conductor. Thus

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ProblemDetermine the magnetic field at the center of a circular loop of radius R carrying current I.

where ds = R µ andThe magnet field of a small current loop is just like that of a small bar magnet with B lines emerging from an imaginary north pole and looping back to end in an imaginary south pole. Thus the field of a small current carrying loop is that of a magnetic dipole. See figure on the next slide.

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Ampere‘s LawA magnet field proportional to the current is generated and follows the right hand rule. The current is represented by the curl of the fingers and the thumb points in the direction of the north pole. The integral form is written

Ampere‘s Law is only useful for determining the magnet field for simple cases such as when one assumes that the magnet field of an inductive coil is over all homogenous which is only the case for an infinitely long coil.

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If we consider an ideal coil with N windings per meter, we can insert a rectangular frame through the coil. The top side of the coil lies on the same axis as the length of the coil and the right and left sides are infinitely long. The magnetic field is assumed to lie perpendicular to the sides, therefore the components in the direction of the frame are zero. The other side is infinitely far away so the magnet field must be zero there.

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Consider the line integral shown in the figure. The path integral of the magnetic field around this integration path is equal to

where L is the horizontal length of the integration path. The current enclosed by the integration path is equal to

where N is the number of turns enclosed by the integration path and I0 is the current in each turn of the solenoid. Using Ampere's law we conclude that

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or

where n is the number of turns of the solenoid per unit length. The equation shows that the magnetic field B is independent of the position inside the solenoid. We conclude that the magnetic field inside an ideal solenoid is uniform.

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Problem The figure below shows two views of a circular loop of radius 4 cm placed within a uniform magnetic field B of magnitude 0.3 T. a.) What is the magnetic flux through the loop?

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b.) What would be the magnetic flux through the loop if the loop were rotated ?

c.) What would be the magnetic flux through the loop if the loop were rotated ?

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a.) Since B is parallel to A, the magnetic flux is equal to BA:

b.) Since the angle between B and A is , the magnetic flux through the loop is

c.) If the angle between B and A is , the magnetic fluxthrough the loop is zero since .

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Dynamo on your Bicycle A dynamo works to light up your bike using induction. When you pedal your bike, you rotate a magnet in an iron core and produce a current in an inductor coil.