The current direction is into the page at points A and B and out of the page at points C and D. Calculate the magnitude and direction of the magnetic field at point P, located at the center of the square of edge length 0.2m. Solution:
At point P, (the origin of the chosen coordinate system) wires A and D each produce a magnetic field of magnitude m0I/(2pd) pointing towards wire B. Here d2 = 0.12+0.12. Wires C and B each produce a magnetic field of magnitude m0I/(2pd) pointing towards wire D. The y-components of all the fields add, while the x-components cancel. The total field at point P therefore has magnitude
4m0Isin(45o)/(2pd) = 80p10-7sin(45o)/(2p(0.2)1/2)T = 20mT
and points into the negative y-direction.
Maxwell's equations can be used to derive the Biot-Savart law. The Biot-Savart law can be used to find the magnetic field produced by any distribution of steady currents. For a small segment of wire of length dl carrying a current I, the Biot-Savart law states that the magnetic field dB produced by that segment a distance r from the segment is given by
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Here km is a constant, km = 10-7Tm/A. Often km is written as km = m0/4p, where m0 is the permeability of free space.
The Biot-Savart law is an inverse square law. The directional aspects are such that the magnetic field produced by a current element dl at any point P encircles the straight line passing through dl.
Since steady currents always flow in closed loops, we need to integrate dB over the entire circuit to evaluate the net field B at point P. (This is a vector integral. The contributions dB from different sections add vectorially.) All sections of the loop contribute to B. But because of the inverse-square dependence on distance, the sections closest to P make the largest contributions.
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Some results:
The magnetic field on the axis of a current loop of radius R, a distance z from the center of the loop is
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At the center of the loop z = 0 and B = (m0I/(2R))k.
A small current loop has a dipole moment m = IAn. The magnetic field lines of a loop with dipole moment m are shown below. The field-line pattern is that of a small magnet. We say that m points from the south pole to the north pole of the magnet. (Magnetic field lines therefore exit a magnet at the north pole and enter at the south pole.)
The picture below shows the field pattern in a plane perpendicular to the loop, revealed with iron filings.
The magnitude of the field inside a tightly wound solenoid (away from the ends) with n = N/l turns per unit length is B = m0nI. The direction of the field is given by the right-hand rule. Curl the fingers of your right hand in the direction that the current flows in the solenoid. Your thumb points in the direction of B.