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PHY110W MAGNETISM <strong>MAGNETIC</strong> <strong>FIELDS</strong> <strong>DUE</strong> <strong>TO</strong> <strong>CURRENTS</strong><br />
<strong>MAGNETIC</strong> FIELD <strong>DUE</strong> <strong>TO</strong> CURRENT<br />
IN A LONG, STRAIGHT WIRE<br />
d s <br />
i<br />
s<br />
θ<br />
R<br />
r<br />
× dB<br />
P<br />
Using symmetry (every current element ds in the upper<br />
half of the wire has a corresponding element in the<br />
lower half causing the same field at P), the magnitude<br />
of the magnetic field at point P due to a long, straight,<br />
current-carrying wire is<br />
∞<br />
0i<br />
2 4<br />
µ<br />
B = sinθ<br />
ds<br />
π ∫ (30-7)<br />
2<br />
r<br />
0<br />
= ° − =<br />
R<br />
, and<br />
r<br />
where sinθ<br />
sin( 180 θ)<br />
∴ B =<br />
µ 0i<br />
2π<br />
∞<br />
∫<br />
0<br />
R<br />
(<br />
2 2)<br />
s<br />
+<br />
R<br />
32<br />
ds<br />
2 2<br />
r = s + R<br />
∞<br />
µ 0i<br />
⎡ s ⎤<br />
∴ B = 12<br />
2π<br />
R ⎢( 2 2<br />
s R ) ⎥<br />
⎣ + ⎦0<br />
µ 0i<br />
∴ B = (30-6)<br />
2π<br />
R<br />
The direction of B is given by the RH curled fingers rule.<br />
14
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