Simple Nature - Light and Matter

is very much as it was before, except that all the field vectors havehad one unit worth of ŷ added to them. But what do we get if wetake the curl of −xŷ + ŷ? The curl, like any god-fearing derivativeoperation, has the additive propertysocurl (F + G) = curl F + curl G ,curl (−xŷ + ŷ) = curl (−xŷ) + curl (ŷ) .But the second term is zero, so we get the same result as at theorigin.A field that goes in a circle example 15⊲ What is the curl of the field xŷ − y ˆx?⊲ Using the linearity of the curl, and recognizing each of the termsas one whose curl we have already computed, we find that thisfield’s curl is a constant 2ẑ. This agrees with the right-hand rule.The field inside a long, straight wire example 16⊲ What is the magnetic field inside a long, straight wire in whichthe current density is j?⊲ Let the wire be along the z axis, so j = jẑ. Ampère’s law givescurl B = 4πkc 2 jẑ .In other words, we need a magnetic field whose curl is a constant.We’ve encountered several fields with constant curls, but the onlyone that has the same symmetry as the cylindrical wire is xŷ−y ˆx,so the answer must be this field or some constant multiplied by it,The curl of this field is 2bẑ, soand thusB = b (xŷ − y ˆx) .2b = 4πkc 2 j ,B = 2πk j (xŷ − y ˆx) .c2 The curl in component formNow consider the fieldF x = ax + by + cF y = dx + ey + f ,Section 11.4 Ampère’s Law In Differential Form (Optional) 681