Question

Problem 1 Three long straight wires are parallel to each other and carry currents directed into or out of the page, as shown. Wires 1 and 2 carry a current I, and wire 3 carries a current 4. (Note: Be sure to use unit vector notation in each part) 2 2d A) Find an expression for the net magnetic field B at the origin. (10 pts.) B) Find the force on a charged particle q1.0uC passing through the origin with speed 1.0 m /s directed out of the page ( v-1.0k m/s). Assume that I = 1.0 A and d 1.0 m. (5 pts.) C) Find the force on a fourth wire with length 10 m located at the origin, parallel to the other wires, and carrying a current I directed out of the page. (5 pts.) Bonus:Where could you place the fourth wire from part c so that the magnetic field at the origin is zero? (3 pts.)

Answer 1

a) Magnetic filed at origin due to wire 1,2 and 3 is given by,

$\vec{B}=\vec{B_{1}}+\vec{B_{2}}+\vec{B_{3}}=\frac{\mu_{0}}{2\pi}\left [\left ( \frac{I_{1}}{r_{1}} \right )\hat{y}+\left ( \frac{I_{2}}{r_{2}} \right )\hat{y}+\left ( \frac{I_{3}}{r_{3}} \right )\hat{x} \right ]$

$\vec{B}=\frac{\mu_{0}}{2\pi} \left ( \frac{2I}{d}\hat{y}+\frac{4I}{2d}\hat{x} \right )$

$\vec{B}=\frac{\mu_{0}I}{\pi d}\left [\hat{y} +\hat{x}\right ]$

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b) Force,

$\vec{F}=q(\vec{v}\times \vec{B})=$$\vec{F}=q(\vec{v}\times \vec{B})=\frac{\mu_{0}qI}{\pi d}v\hat{z}\times(\hat{x}+\hat{y})=\frac{\mu_{0}qvI}{\pi d}(\hat{y}-\hat{x})$

$\vec{F}=\frac{4\pi\times 10^{-7}\times10^{-6}\times 1\times1}{\pi}(\hat{y}-\hat{x})=\left (4\times 10^{-13}N \right )(\hat{y}-\hat{x})$

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c) Force on the fourth wire,

$\vec{F}=IL\hat{z}\times\frac{\mu_{0}I}{\pi d}(\hat{y}+\hat{x})=\frac{\mu_{0}I^{2}L}{\pi d}(\hat{y}-\hat{x})$

$\vec{F}=\frac{4\pi\times 10^{-7}\times10}{\pi }(\hat{y}-\hat{x})=\left (4\times 10^{-6}N \right )(\hat{y}-\hat{x})$