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Q.

Two positive charges q₁ and q₂are moving with velocities v₁ and v₂ when they are at points A and B, respectively, as shown in figure. The magnetic force experienced by charge q₁ due to the other charge q₂ is

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a

$\frac{μ_{0} q_{1} q_{2} v_{1} v_{2}}{2 \sqrt{2} π a^{2}}$

b

$\frac{μ_{0} q_{1} q_{2} v_{1} v_{2}}{\sqrt{2} π a^{2}}$

c

$\frac{μ_{0} q_{1} q_{2} v_{1} v_{2}}{4 \sqrt{2} π a^{2}}$

d

$\frac{μ_{0} q_{1} q_{2} v_{1} v_{2}}{8 \sqrt{2} π a^{2}}$

answer is A.

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Detailed Solution

$Magnetic field at A due to q_{2} : {\vec{B}}_{2} = \frac{μ_{0}}{4 π} \frac{q_{2} v_{2} r \sin 45 °}{r^{3}} \hat{k} where r = \sqrt{2} a \Rightarrow {\vec{B}}_{2} = \frac{μ_{0}}{4 π} \frac{q_{2} v_{2}}{2 \sqrt{2} a^{2}} \hat{k} Force on q_{1} : \vec{F} = q_{1} {\vec{v}}_{1} \times {\vec{B}}_{2} = \frac{μ_{0}}{4 π} \frac{q_{1} q_{2} v_{1} v_{2}}{2 \sqrt{2} a^{2}} (- \hat{j})$

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Two positive charges q1 and q2 are moving with velocities v1 and v2 when they are at points A and B, respectively, as shown in figure. The magnetic force experienced by charge q1 due to the other charge q2 is