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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 CIRCULAR WIRE<br />
i<br />
R<br />
α<br />
x<br />
r <br />
dB y<br />
P<br />
α<br />
dB <br />
dB x<br />
From the Biot-Savart law, the field at point P (along the<br />
axis of the a circular loop, radius R), due to a differential<br />
current element ids at the top of the loop, is given by<br />
<br />
<br />
µ 0 idssinα<br />
µ 0i<br />
ds<br />
dB = =<br />
2<br />
( 2 2<br />
4π<br />
r 4π<br />
R + x )<br />
By symmetry, all the components perpendicular to the<br />
axis (dB y ) sum to zero, so we need consider only dB x :<br />
dB<br />
x<br />
µ 0i<br />
ds µ 0i<br />
ds R<br />
= cosα<br />
=<br />
4π<br />
R + x 4 R + x R + x<br />
( 2 2) π ( 2 2) ( 2 2)<br />
12<br />
i, R and x have the same values for all elements around<br />
the loop, so when we integrate around the circumference…<br />
B<br />
x<br />
µ 0iR<br />
=<br />
4π<br />
R + x<br />
(<br />
2 2)<br />
32<br />
∫ ds (and ∫ ds is just 2πR)<br />
so<br />
B<br />
x<br />
=<br />
2<br />
µ iR<br />
0<br />
2<br />
(<br />
2 2)<br />
R<br />
+<br />
x<br />
32<br />
(30-28)<br />
16
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