Application of the Magnetic Field Equation

Application of the Magnetic Field Equation - limits of...

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Application of the Magnetic Field Equation This equation is quite complicated, and is difficult to understand on a theoretical level. Thus, to show its applicability, we will use the equation to calculate something we already know: the field from a straight wire. We begin by drawing a diagram showing a straight wire, including an element dl , in relation to a point a distance x from the wire: Figure %: An element dl on a long wire, contributing to the magnetic field at P , a distance x from the wire From the figure, we see that the distance between dl and P is . In addition, the angle between and dl is given by sin θ = . Thus we have the necessary values to plug into our equation: dB = = Now that we have an expression for the contribution of a small piece, we may sum over the whole wire to find the total magnetic field. We integrate our expression with respect to l , with
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Unformatted text preview: limits of integration from to - : B = dB = = = Since I , x and c are constants, we may remove them from the integral, simplifying the calculus. This integral is still quite complicated, and we must use a table of integration to solve it. It turns out that the integral is equal to . We evaluate this expression using our limits: B = When we plug infinity into our expression we find that l , implying that plugging in a value of infinity yields the value 1/ x 2 . When we plug in our negative infinity, we get -1/ x 2 in a similar manner. Thus: B = - = This is the equation we saw earlier for the field of a straight wire, implying that our calculus equation derived earlier is correct. The math that accompanies this kind of calculation is difficult, and rarely used, but it is essential for deriving the formulae we will encounter in the next section ....
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Application of the Magnetic Field Equation - limits of...

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