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Question

In the figure, two long straight wires are perpendicular to the page and separated by distance $$d_{1}=0.75\hspace{0.05cm} cm$$. Wire $$1$$ carries $$6.5 A$$ into the page. What are the (a) magnitude and (b) direction(into or out of the page) of the current in wire $$2$$ if the net magnetic field due to the two currents is zero at point P located at distance $$d_{2}=1.50\hspace{0.05cm}cm$$ from wire $$2$$?
1774629_6bd8f81a10dd4a558239bb2dc643735e.png


Solution

(a)$$B_{P_{1}}=\mu_{0}i_{1}/2\pi r_{1}$$ where $$i_{1}=6.5\hspace{0.05cm}A$$ and $$r_{1}=d_{1}+d_{2}=0.75\hspace{0.05cm}cm+1.5\hspace{0.05cm}cm=2.25\hspace{0.05cm}cm$$, 

and $$B_{P_{2}}=\mu_{0}i_{2}/2\pi r_{2}$$ where $$r_{2}=d_{2}=1.5\hspace{0.05cm}cm$$. 

Magnetic field at point P is zero, $$B_{P_{1}}=B_{P_{2}}$$ we get $$i_2=i_1\bigg( \dfrac{r_{2}}{r_{1}}\bigg)=(6.5\hspace{0.05cm}A)\bigg( \dfrac{1.5\hspace{0.05cm}cm}{2.25\hspace{0.05cm}cm} \bigg)=4.3\hspace{0.05cm}A$$
(b) Using the right-hand rule, we see that the current $$i_2$$ carried by wire 2 must be out of the page.

Physics
NCERT
Standard XII

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