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| Summary |
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- The Biot-Savart law asserts that the magnetic field dB due to an element dl carrying a steady current i at a point P at a distance r from the current element is:
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- The constant m0 has the exact value
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m0 = 4p x 10-7 T m A-1 |
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| It is the permeability of free space. It is related to e0, the permittivity of free space and the speed of light in free
space c, by  |
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- The magnitude of the magnetic field at a distance R from a long, straight wire carrying a current i is given by:
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| The field lines are circles concentric with the wire. |
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- The magnitude of the magnetic field due to a N-turn circular coil of radius R carrying a current i at an axial distance x from the center is
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At the center this reduces to  |
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- Ampere's Circuital Law: We have discussed a simplified form of this law. If B is directed along the tangent to every point on the perimeter L of a closed curve and is constant in magnetic along perimeter then.
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| BL = m0 ie |
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| where ie is the net, current enclosed by the closed circuit. |
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- Employing Ampere's law one can show that the magnitude of the field B inside a long solenoid carrying a current i is
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| B = m0 ni |
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| where n is the number of turns per unit length. For a toroid one obtains, |
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| where N is the total number of turns and r the average radius. |
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- The total force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E, respectively is called the Lorentz force. It is given by the expression:
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| The magnetic force q (v x B) is normal to v and work done by it is zero. |
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- In a uniform magnetic field B, the charge q described in (7) executes circular orbit in the plane normal to B. Its frequency of uniform circular motion is called the cyclotron frequency and is given by:
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| This frequency is independent of the particle's speed and radius. This fact is exploited in a machine, the cyclotron,
which is used to accelerate charged particles. |
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- A straight conductor of length l and carrying steady current i experiences a force F in a uniform external magnetic field B,
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| F = il x B |
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| where |l| = l and the direction of l is given by the direction of the current. |
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- Parallel currents attract and anti-parallel currents repel.
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- A planar loop carrying a current i, having N closely, wound turns, and an area A possesses a magnetic moment m where, m = N i A
and the and the direction of m is given by the right-hand thumb
rule: curl the palm of your right hand along the loop with the fingers
pointing in the direction of the current. The thumb sticking out gives
the direction of m (and A).
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| When this loop is placed in a uniform magnetic field B, the force F on it is |
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| F = 0 |
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| And the torque on it is, |
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| t = m x B |
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| In a moving coil galvanometer, this torque is balanced by a counter - torque due to a spring, yielding. |
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| kf = Ni AB |
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| where f is the equilibrium deflection and k the torsion constant of the spring. |
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- An electron moving around the central nucleus has a magnetic moment
ml given by:
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| where l is the magnitude of the angular momentum of the circulating electron about the central nucleus. The
smallest value of ml is called the Bohr magneton mB and it is |
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| mB = 9.27 x 10-24J/T |
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