Question 7.2: Find H at (-3, 4, 0) due to the current filament shown in Fi...

Let H=H_{1}+H_{2}, where H_{1} and H_{2} are the contributions to the magnetic field intensity at P(-3, 4, 0) due to the portions of the filament along x and z, respectively.

H_{2}=\frac{I}{4\pi\rho}(\cos\alpha_{2}-\cos\alpha_{1})a_{\phi}

At P(-3, 4, 0),  \rho=(9+16)^{1/2}=5,  \alpha_{1}=90^{\circ},  \alpha_{2}=0^{\circ}, and a_{\phi} is obtained as a unit vector along the circular path through P on plane z=0 as in Figure 7.7(b).

The direction of a_{\phi} is determined using the right-handed-screw rule or the right-hand rule. From the geometry in Figure 7.7(b)

a_{\phi }=\sin\theta a_{x}+\cos\theta a_{y}=\frac{4}{5}a_{x}+\frac{3}{5}a_{y}

Alternatively, we can determine a_{\phi} from eq (7.15).

a_{\phi }=a_{\ell}\times a_{\rho}

At point P, a_{\ell} and a_{\rho} are as illustrated in Figure 7.7(a) for H_{2}. Hence

a_{\phi }=-a_{z}\times\left(-\frac{3}{5}a_{x}+\frac{4}{5}a_{y}\right)=\frac{4}{5}a_{x}+\frac{3}{5}a_{y}

as obtained before. Thus

H_{2}=\frac{3}{4\pi(5)}(1-0)\frac{(4a_{x}+3a_{y})}{5}=38.2a_{x}+28.65a_{y}mA/m

It should be noted that in this case a_{\phi} happens to be the negative of the regular a_{\phi} of cylindrical coordinates. H_{2} could have also been obtained in cylindrical coordinates as

H_{2}=\frac{3}{4\pi(5)}(1-0)(-a_{\phi})=-47.75a_{\phi}mA/m

Similarly, forH_{1} at P

\rho=4,  \alpha_{2}=0^{\circ},  \cos \alpha_{1}=3/5

and

a_{\phi}=a_{z}   or   a_{\phi}=a_{\ell}\times a_{\rho}=a_{x}\times a_{y}=a_{z}

Hence

H_{1}=\frac{3}{4\pi(4)}(1-\frac{3}{5})a_{z}=23.87a_{z}mA/m

Thus

H=H_{1}+H_{2}=38.2a_{x}+28.65a_{y}+23.87a_{z}mA/m

or

H=-47.75a_{\phi}+23.87a_{z}mA/m

Notice that although the current filaments appear to be semi-infinite (they occupy the positive z- and x-axes), it is only the filament along the z-axis that is semi infinite with respect to point P. Thus H_{2} could have been found by using eq. (7.13)

H=\frac{I}{4\pi\rho}a_{\phi}

but the equation could not have been used to find H_{1} because the filament along the x-axis is not semi-infinite with respect to P.