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Question 5.19: A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west. The earth’s magnetic field at the place is 0.39 G, and the angle of dip is 35º. The magnetic declination is nearly zero. What are the resultant magnetic fields at points 4.0 cm below the cable?Class 12 - Physics - Magnetism And Matter Magnetism And Matter Page-202

Answer»

Let us first decide the directions which can best represent the situation.
angle of dip , \delta=35°
Earth's MAGNETIC FIELD, B = 0.39G
horizontal component of earth's magnetic field,B_H=Bcos\delta=0.39cos35°
= 0.32G

similarly, vertical component of earth's magnetic field, B_v=Bsin\delta
=0.39sin35° = 0.22G

telephone cable carry a total current of 4A in direction east to west . we want RESULTANT magnetic field 4cm below as shown in figure.
we have to find magnetic field DUE to wire ,
B_{wire}=\frac{\mu_0}{4\pi}\frac{2I}{r}\\\\=10^{-7}\times\frac{2\times4}{4\times10^{-2}}\\\\=2\times10^{-5}T=0.2G

now, see figure,
net magnetic field,B_{net}=\sqrt{(B_H-B_{wire})^2+B_v^2}\\\\=\sqrt{(0.32-0.2)^2+0.22^2}=0.25G


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