Question
Do magnetic forces obey Newton's third law. Verify for two current elements $\text{dI}_1=\text{dI}\hat{\text{i}}$ located at the origin and $\text{dI}_2=\text{dI}\hat{\text{j}}$ located at $(0, R, 0)$. Both carry current $I.$
✓
Answer
Key concept: In this problem first we have to find the direction of magnetic field due to one wire at the point on other wire, then the magnetic force on that current carrying wire.
According to Biot$-$Savart's law, magnetic field $B$ is parallel to $idl \times r$ and $idl$ is the current carrying element having it's direction along the direction of flow of current.
Here, for the direction of magnetic field, at $dl_2$, located at $(0, R, 0)$ due to wire dlx is given by $B \| idl \times r$ or $i \times j ($because point $(0, R, 0)$ lies ony$-$axis$)$, but $i \times j = k$.
So, the direction of magnetic field at $dl_2$ is along $z-$direction.
The direction of magnetic force exerted at $dl_2$ due to the magnetic field of first wire is along the $x-$axis.
$F - i(l \times B),$ i.e., $F \| (i \times k)$ or along $-j$ direction.
Therefore, force due to dl $[$on $dl_2$ is non$-$zero.$]$
Now, for the direction of magnetic field, at $dx$, located at $(0, 0, 0)$ due to wire $d_2$ is given by $B \| idl \times r$ or $j \times -j ($because origin lies on $y-$direction $w.r.t$. point $(0, R, 0)$, but $j \times -j = 0.$
So, the magnetic field at dx does not exist.
Force due to $dl_2$ on $dl_1$, is zero.
So, magnetic forces do not obey Newton's third law. But they obey Newton's third law if current carrying element are placed parallel to each other.
According to Biot$-$Savart's law, magnetic field $B$ is parallel to $idl \times r$ and $idl$ is the current carrying element having it's direction along the direction of flow of current.
Here, for the direction of magnetic field, at $dl_2$, located at $(0, R, 0)$ due to wire dlx is given by $B \| idl \times r$ or $i \times j ($because point $(0, R, 0)$ lies ony$-$axis$)$, but $i \times j = k$.
So, the direction of magnetic field at $dl_2$ is along $z-$direction.
The direction of magnetic force exerted at $dl_2$ due to the magnetic field of first wire is along the $x-$axis.
$F - i(l \times B),$ i.e., $F \| (i \times k)$ or along $-j$ direction.
Therefore, force due to dl $[$on $dl_2$ is non$-$zero.$]$
Now, for the direction of magnetic field, at $dx$, located at $(0, 0, 0)$ due to wire $d_2$ is given by $B \| idl \times r$ or $j \times -j ($because origin lies on $y-$direction $w.r.t$. point $(0, R, 0)$, but $j \times -j = 0.$
So, the magnetic field at dx does not exist.
Force due to $dl_2$ on $dl_1$, is zero.
So, magnetic forces do not obey Newton's third law. But they obey Newton's third law if current carrying element are placed parallel to each other.
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