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Q)

Do magnetic forces obey Newton's third law : Verify for two current elements $\overrightarrow{dl_1}= dl \hat i$ located at the origin and $\overrightarrow{dl_2}= dl \hat j$ located at $(0,R,0). Both carry current I.

$\begin{array}{1 1} \text{yes they do obey} \\ \text{No, they do not obey} \\ \text{can not be determined } \\ \text{None of the above} \end{array} $

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A)
Solution :
Magnetic field due to current element $I \overrightarrow{dl_1}$ at a point on element $I \overrightarrow{dl_2}$
Force on $I \overrightarrow{dl_2}$ is $I \overrightarrow{F}= Id_1 \hat j \times I \overrightarrow{B}$
$\quad= Idl \hat j \times \large\frac{\mu_0}{4 \pi} \frac{Idl}{R^2} $$\hat k$
$\quad = \large\frac{\mu_0}{4 \pi} \frac{I^2 dl^2}{R^2}$$(j \times k)$
$\quad = \large\frac{\mu _0}{4 \pi} \frac{I^2 dl^2}{R^2}$$( \hat j \times \hat k) \neq 0$
Force on $I \overrightarrow{dl_1}$ due to magnetic field by current $I \overrightarrow{dl_2}$ is zero because the magnetic field of $I \overrightarrow{dl_2}$ has no interaction with current element $I \overrightarrow{dl_1}$
Magnetic forces on current element in this case do not obey newton's third law
Answer : $\text{No, they do not obey}$
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